On the conjecture of groups with the same number of centralizers

For any group [Formula: see text], let [Formula: see text] denote the set of all centralizers of [Formula: see text]. K. Khoramshahi and M. Zarrin [Groups with the same number of centralizers, J. Algebra Appl. 20 (2021) 2150012], posed the following conjecture: Let [Formula: see text] and [Formula: see text] be finite groups. Is it true that if [Formula: see text] and [Formula: see text], then [Formula: see text] is isoclonic to [Formula: see text]? In this paper, among other things, we give a negative answer to this conjecture.