  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.
