According to Wikipedia, a $CIT$-group is a group such that the centralizer of any involution is a 2-subgroup. The structure of these groups is known from the works of Suzuki and others.
Here is my question: has the odd case also been studied?
For example, if we define a $CIT_p$-group a group such that the centralizer of any $p$-element is a p-subgroup, what is known about these groups?
Any reference would be highly appreciated.