Automorphism groups of supergraphs of the power graph of a finite group

For a finite group GG, the power graphP(G)P(G) is a graph with the vertex set GG, in which two distinct elements are adjacent if one is a power of the other. Feng, Ma and Wang (Feng et al., 2016) described the full automorphism group of P(G)P(G). In this paper, we study automorphism groups of the main supergraphs and cyclic graphs, which are supergraphs of P(G)P(G). It is proved that the automorphism group of these graphs can be written as a combination of Cartesian and wreath products of some symmetric groups. The full automorphism groups of these graphs of certain finite groups are also calculated.