‘And as they spoke to the people, the priests and the captain of the temple and the Sadducees came upon them, being sore troubled because they taught the people, and proclaimed in Jesus the resurrection from the dead.'
The preaching of Peter was raising eyebrows among the authorities in the Temple. It may well be that they had been willing to overlook his sermon at Pentecost because, like some of the crowd, they simply thought that he was drunk, and that it was not too serious and would not happen again. It had after all resulted from a rather unusual and inexplicable situation.
However, now that it had happened a second time they could not overlook it and felt that it was therefore necessary to examine the matter and if necessary give an official admonition. Such goings on could not be allowed in the Temple. The thing that caused most offence to the Temple authorities themselves was Peter's teaching on the resurrection of the dead. While the Pharisees believed in the resurrection of the dead, the Sadducees, including the chief priests, most decidedly did not. And it was they who had overall responsibility for the Temple. So when Peter began teaching about the resurrection of the dead, and proclaiming that God would intervene in world affairs, they took offence.
It will be noted that those who gathered against them were all Sadducees. The priests and other Sadducees (most fairly rich and important) probably reported what they had heard to the captain of the Temple (a senior chief priest responsible for maintaining order and reverence in the temple, or one of his deputies), who then came with them in order to deal with these troublemakers. They clearly felt that their prerogatives were being trodden on. It was recognised that the resurrection from the dead might be taught in the synagogues (by the Pharisees), but not, if they could help it, in the Temple by any wandering preacher.
In fact the Sadducees would not have liked the whole tenor of the Apostolic teaching for the Sadducees also denied the principle of divine action in the world and wanted to maintain the status quo. Furthermore they still had vividly in their minds the way in which this Jesus in Whose Name these men were acting had attacked the sources of their profits in the trading that took place in the Temple.