On the rings generated by the inner automorphisms of finite groups

For a finite group G, let I(G) denote the set of all finite sums of inner automorphisms of G. When I(G) forms a ring, G is referred to as an I-group. It is known that if G is an I-group, then it is nilpotent of class at most 3, and that I(G) is a commutative ring if and only if G is nilpotent of class at most 2. We characterize the ring I(G) for an I-group G. Additionally, for cases where I(G) is a commutative ring and G is of order pⁿ (with p being a prime and n = 3 or 4), as well as for orders 3⁵ and 3⁶, we determine the ring structure of I(G).