Recognition of almost quasisimple groups using involution counts

Prove that the group order n(G), the number i(G) of involutions, and the indicator tuple δ^{eo}(G) of element orders form a full system of invariants for all but finitely many almost quasisimple groups.

Background

The paper gives examples showing that order together with element-order data does not distinguish every almost simple or quasisimple group related to a simple group. It nevertheless proposes that adding the number of involutions should suffice for all but finitely many almost quasisimple groups.

The conjecture is motivated by the observed near-sufficiency of the group order and element-order set for recognizing groups related to simple groups.