A sick person says he cannot attend the prayers in the mosque and alternates.

He or his sons lead the prayer in congregation for every obligatory prayer, so is this action permissible for him since they left the mosque prayer?

No, this is a mistake. Just because he is sick and cannot come to the mosque does not mean his sons are prevented from going to the mosque under the excuse of "come pray with me."

No, they should go and pray in the mosque. When they finish, one of them can come and pray with him in congregation.

Or before he goes to the mosque. The important thing is,

The one who is excused is the sick person. As for the rest of the sons, they do not pray at home using the excuse that their father is sick, unless he needs one of them to stay with him. That is a different matter.

But just for the sake of praying in congregation, no, they should pray in congregation at the mosque, then come, all of them or some of them, and pray with you.

A second congregation, but they should not leave the mosque congregation because their father.

Maryam, we ask Allah, the Blessed and Exalted, to grant her healing and well-being.