What was Peter hoping Israel would realize?
The hope was that Israel would wake up and acknowledge that the One they killed was the Promised Messiah, the One promised according to the covenants. Despite their ignorance, Peter longed for the nation to believe in Jesus as Messiah.
Les explains that Peter was still coming back to Israel, hoping against hope that the nation would wake up and realize that the One they killed was the Promised Messiah. The message had to go to the Jew first, then the Gentile, but Israel would not believe. Peter addresses them as brethren, not Gentiles, and says they acted in ignorance, as did their rulers. Les notes this matches Christ's prayer from the cross, 'Father forgive them, for they know not what they do,' and Stephen's similar statement in chapter 7.