Inter are moving on with keeping Inzaghi in the manager position. I think they are doing the right thing as well. Because I call their last season performance successful. They won two titles like Coppa Italia and Italian Super Cup first of all. After that they qualified for the Champions League finale and made quite big difficulties for Manchester City there as well. Maybe they weren't in a title race in the Serie A. But there was a reason for that as well. They couldn't start the season well and it was late when they started to do better.
They had defensive issues at the first times of that season. They should make sure now that they don't have the same experience this season.
I didn't follow the manager discussion for Inter Milan recently, but have there been discussions about Inzaghi for real? Or did Inzaghi hesitate and think about leaving the club? The club has absolutely no reason to sack him nor have discussions about him. Actually, I would rather expect the club to raise the incentives for Inzaghi to stay for longer and try even harder. Though I could see Inzaghi having some doubts whether he will be able to repeat or even come close to what he achieved last season. The Champions League final is unrealistic, but they should make the Serie A title the priority number 1.