Apostasy is an issue that has caused debate for quite sometime. In the end, the answer does should not apply to any people today since there is no Islamic state to uphold the judgement. And anyone who is carrying our Shari'ah criminal punishment is doing so against the Islamic law, since that would require an Islamic state with an established court. That being said, let's see how well I can answer this.
There are two types of apostasy in Islam. Simply Apostasy and Complete Apostasy. Let's look first at the two different definitions of these terms.
Simple Apostasy = No longer wishing to follow the Muslim faith and simply wishing to practice a separate religion other than Islam.
Complete Apostasy = Rejecting the faith of Islam while simultaneously attempting to undermine the Islamic state via violence or civil disorder by joining in the enemies of Islam in the sense of waging war on Muslims.
There is a consensus among scholars that complete apostates should be put to death. It basically equals the treason law that many countries follow. It is when you get to the idea of execution for simple apostasy you start seeing the differences of opinion.
It is the consensus of all four schools of thought within the Sunni tradition that apostates should be executed, however the argument comes from the question if these apostates are simple or complete apostates. Many people who support the execution law will quote the founding scholars of the four schools of thought saying that apostasy should be punished with execution, however, I have not seen any evidence explaining which types of apostates they are referring too. It also must be noted that some of the schools of thought traditionally supported the idea of fallibility among scholars and that fatwas were not the "rulings of God" but rather the opinions of those scholars. Traditionally the Hanafi school of though, the largest school of thought, even stated that fatwas be limited to specific times and places according to Islamic scholar Hamza Yusuf.
Bearing all of this in mind let us look at the Quran. Apostasy is mentioned 26 times in the Quran but yet there is no earthly punishment mentioned within the Quran. Muhammad himself was confronted with multiple apostates during his lifetime and there is no recorded evidence of him putting those apostates to death. Rather, he simply let them leave as long as they were peaceful about it.
There are also many quotes from the Quran that agree with this stance:
There is no compulsion in religion. Right has become distinct from wrong. So whoever rejects evil and puts faith in God has grasped the most trustworthy hand-hold that never breaks. And God is hearing, knowing. (2:256)
The apostasy law is first thought to gain large support during a time of struggle and religious upheaval within the time of the Islamic state. Bearing that in mind it's easy to see why the four schools of thought came down so hard on this decision, it was created as a way of protecting the state and religion during that time.
It is my firm belief from the research that I have done that there is no call for execution for simple apostasy within Islam. And, unfortunately, the apostasy law has been twisted to encompass people outside of it's original meaning.
Allow me to be clear, this seems to be a minority opinion based on the people I have dealt with. But I hope that it provides some insight into the debate. Since I am a new user, I can only post two links within my sources. I've provided the two that I feel best present my argument. One is a detailed look at the apostasy law, and the other is a video explaining how scholarly issued fatwas can become twisted into the dichotomy of the religion. I hope that you find the satisfactory.