The only mention of this group in the Book of Acts is in Acts 15:5. It says, when Paul came to Jerusalem in order to defend himself for relaxing Mosaic law for Gentile converts, that some of the Pharisaic party raised objections, insisting that Paul's converts be circumcised, and observe Mosaic law.
There seems to have been some members of the Council of Elders who were former Pharisees. They were probably the poorer members of the Pharisaic party, who were attracted by Yeshua`'s desire to unite the squabbling Jewish religious parties and return to the ways of YHWH. Because of their strict adherence to the Torah, and their piety, I have called this group Pietists.
Throughout history, Christians have insisted that the break between Christianity and Judaism was caused by so-called "Jewish Christians'" insistance on Gentile converts following Mosaic law, and that men be circumcised.
One has to ask the question: If they were not following Torah, and were not being circumcised, how could they be considered Jewish in any way? If anyone enters a religion, one would expect that those converts adhere to the minumum requirements of that religion in order to be considered members. One cannot simply say that you are a Jew, and then not do anything Jewish!
Consider this: Jesus was born, raised, and died a Jew; ALL the Emissaries (Apostles) were born Jewish, and taught a form of Judaism according to their master's teaching; Yeshua`'s first followers in the Galilee were Jewish. Yeshua` never taught that his group was leaving Judaism, so there is no reason to assume that they were teaching anything but Judaism.
I think that the reason why circumcision became such an issue for the Pietists, was because there was nothing else that Christians believed or taught that would qualify them to be called Jewish. The issue was not circumcision, but rather, "What religion exactly are you trying or pretending to belong to? Either follow the club rules required for membership, or stop falsely claiming that you are Followers of the Way!
It would not be honest to try to guess at what this group believed. All we can surmise is that they would have practised a very strict piety, and adhered closely to the Torah. Since the ordinary Jewish man and woman went to Pharisaic elders for clarification on various legal matters, they may even have retained Pharisaic legal codes and the Oral Law. More than this, it is not possible to make any further guesses without going into the realms of fantasy.