Neurons that send axons into the cerebellum in the middle cerebellar peduncle are controlled by which part of the brain?

If they go to the cerebellum -then by the cerebellum. But the brain connections are complex and not compartmentalized so many areas can be involved

You need to revise your appreciation of the complexity of the human brain (or even the mouse brain). Everything connects with everything else, and everything goes everywhere.

Not clear why you are asking this question as it assumes a simplistic anatomy regarding a very complicated connective interaction. Propose you specify what you wish to understand