The prohibitions of the Sabbath was prescribed for the Bani Israel by Allah.

> [Quran 4:154][1] And We raised over them the mount for [refusal of]
> their covenant; and We said to them, "Enter the gate bowing humbly",
> and **We said to them, "Do not transgress on the sabbath"**, and We took
> from them a solemn covenant.


And those who transgressed it were punished:

> [Quran 2:65][2] And you had already known about **those who transgressed
> among you concerning the sabbath**, and We said to them, "Be apes,
> despised."
> 
> 
> 
> [Quran 4:47][3] O you who were given the Scripture, believe in what We
> have sent down [to Muhammad], confirming that which is with you,
> before We obliterate faces and turn them toward their backs or curse
> them as **We cursed the sabbath-breakers**. And ever is the decree of
> Allah accomplished.


I am not aware of any narration that states that it was only prescribed for a sect among the Jews and that a part of them was exempt. The covenant seems to be from all 12 tribes in the time of Prophet Moses (p.b.u.h). 

The Tafsir of [16:124][4] is that the Sabbath was only prescribed *specifically* for the Jews, and that after the revelation of the Torah to Moses, and that it was not part of the *shariah* of Adam, Noah, Abraham, Isaac, Jacob and Joseph etc. Similarly it is not prescribed for Muslims.

Hadith in [Nisai][5] and [Tirmidhi][6]

> وَعَلَيْكُمْ خَاصَّةً يَهُودُ أَنْ لاَ تَعْدُوا فِي السَّبْتِ
> 
> And for you Jews especially, do not break the Sabbath.

From [Tafsir Qurtubi][7]:

> أي لم يكن في شرع إبراهيم ولا من دينه، بل كان سَمْحا لا تغليظ فيه، وكان
> السبت تغليظاً على اليهود في رفض الأعمال وترك التبسط في المعاش بسبب
> اختلافهم فيه
> 
> 
> This was not in the Shariah and Deen of Ibrahim, rather it was relaxed
> and there was no hardship and strictness in it. The prohibition of the
> Sabbath was on the Jews because an argument arose among them
> about leaving work and seeking of provision.

---

Regarding the details of the argument [ "differed over it" ] which led to the prohibitions, it is narrated in tradition that the Bani Israel were offered Friday as their day of observance, but they argued over it and preferred Saturday, so Saturday was prescribed for them and they were deprived of Friday and that was given to the Muslims.

[Tafsir Ibn Kathir][8] mentions this (see english translation [here][9]):

> ويقال إن الله تعالى شرع ذلك لبني إسرائيل على لسان موسى، فعدلوا عنه،
> واختاروا السبت لأنه اليوم الذي لم يخلق فيه الرب شيئاً من المخلوقات
> الذي كمل خلقها يوم الجمعة، فألزمهم تعالى به في شريعة التوراة، ووصاهم
> أن يتمسكوا به، وأن يحافظوا عليه
> 
> 
> It is narrated that Allah prescribed through Moses's tongue the
> sanctity of Friday for Bani Israel, but they differed about it and
> adopted Saturday and thought that creation was finished on Friday, and
> nothing was created on Saturday. Hence when the Torah was revealed,
> Saturday was prescribed for them and they were commanded to firmly
> observe it. 

[Tafsir Al Qurtubi][7]:
 
> إن موسى عليه السلام أمرهم بيوم الجمعة وعيّنه لهم، وأخبرهم بفضيلته على
> غيره، فناظروه أن السبت أفضل؛ فقال الله له: دعهم وما اختاروه لأنفسهم.
> 
> [A group says] Moses related to them the command for Friday, and gave
> them news of its superiority over other days, but they argued and
> debated with him about the superiority of Saturday, so Allah commanded
> Moses: "Leave them and what they have chosen for themselves".

Hadith in [Sahih Muslim][10]:


>  قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم ‏ "‏ أَضَلَّ اللَّهُ عَنِ
> الْجُمُعَةِ مَنْ كَانَ قَبْلَنَا فَكَانَ لِلْيَهُودِ يَوْمُ السَّبْتِ
> وَكَانَ لِلنَّصَارَى يَوْمُ الأَحَدِ فَجَاءَ اللَّهُ بِنَا فَهَدَانَا
> اللَّهُ لِيَوْمِ الْجُمُعَةِ فَجَعَلَ الْجُمُعَةَ وَالسَّبْتَ
> وَالأَحَدَ وَكَذَلِكَ هُمْ تَبَعٌ لَنَا يَوْمَ الْقِيَامَةِ نَحْنُ
> الآخِرُونَ مِنْ أَهْلِ الدُّنْيَا وَالأَوَّلُونَ يَوْمَ الْقِيَامَةِ
> الْمَقْضِيُّ لَهُمْ قَبْلَ الْخَلاَئِقِ ‏"‏ ‏.‏ وَفِي رِوَايَةِ
> وَاصِلٍ الْمَقْضِيُّ بَيْنَهُمْ ‏
> 
> Messenger of Allah (ﷺ) said: It was Friday from which Allah diverted those who were before us. For
> the Jews (the day set aside for prayer) was Sabt (Saturday), and for
> the Christians it was Sunday. And Allah turned towards us and guided
> us to Friday (as the day of prayer) for us. In fact, He (Allah) made
> Friday, Saturday and Sunday (as days of prayer). In this order would
> they (Jews and Christians) come after us on the Day of Resurrection.
> We are the last of (the Ummahs) among the people in this world and the
> first among the created to be judged on the Day of Resurrection. In
> one narration it is: ', to be judged among them".

  


  [1]: https://quran.com/4/154
  [2]: https://quran.com/2/65
  [3]: https://quran.com/4/47
  [4]: https://quran.com/16/124
  [5]: https://sunnah.com/nasai/37/113
  [6]: https://sunnah.com/tirmidhi/42/46
  [7]: http://www.altafsir.com/Tafasir.asp?tMadhNo=0&tTafsirNo=5&tSoraNo=16&tAyahNo=124&tDisplay=yes&UserProfile=0&LanguageId=1
  [8]: http://www.altafsir.com/Tafasir.asp?tMadhNo=0&tTafsirNo=7&tSoraNo=16&tAyahNo=124&tDisplay=yes&UserProfile=0&LanguageId=1
  [9]: http://www.qtafsir.com/index.php?option=com_content&task=view&id=2927&Itemid=71
  [10]: https://sunnah.com/muslim/7/32