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 ΓH(G) of G is the simple undirected graph with vertex set V(ΓH(G))= (G H)∪ \ 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 ΓH(G) is at most $2$. Moreover, we determine all the subgroups H and the quotient groups G/H such that the cross-cap of ΓH(G) is at most three.