What is the ruling on renting out the personal solar panel that the institution distributed to employees as a trust, knowing that the administration will not object but will not like it?

As long as the institution does not own the plates for the employee, it is considered a borrowed item. The sub-leasing of a borrowed item is a matter of dispute among jurists; some prohibit it because the borrower only owns the right to use, while others permit it because the borrower owns the usufruct.

If the institution grants permission for this, the lease is valid; and the permission must be for a specified period because a lease is not valid unless the duration is defined.

Then, a dispute arises regarding who is entitled to the rent of the sub-leased borrowed item: Is it for the borrower or the lender? The Hanbali school of thought holds that the rent belongs to the lender, and there is no objection to the institution waiving this rent for the questioner.