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 H (G) of G is the simple undirected graph with vertex set V (H (G) ) = (G H) \e\ and two distinct vertices a and b are adjacent if either aH = bᵐ H or bH=aⁿH for some m, n 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 H (G) is at most 2. Moreover, we determine all the subgroups H and the quotient groups GH such that the cross-cap of H (G) is at most three.