Genus and crosscap of normal subgroup based power graphs of finite groups

Let H be a normal subgroup of a group G. The normal subgroup based power graph \(\Gamma _H(G)\) of G is the simple undirected graph with vertex set \(V(\Gamma _H(G))= (G\setminus H)\cup \{e\}\) and two distinct vertices a and b are adjacent if either \(aH = b^m H\) or \(bH=a^nH\) for some \(m,n \in \mathbb {N}\). In this paper, we continue the study of normal subgroup based power graph and characterize all the pairs (G, H), where H is a non-trivial normal subgroup of G, such that the genus of \(\Gamma _H(G)\) is at most 2. Moreover, we determine all the subgroups H and the quotient groups \(\frac{G}{H}\) such that the cross-cap of \(\Gamma _H(G)\) is at most three.