Consider a central extension
$$1 \longrightarrow \mathbb{Z} \longrightarrow G \longrightarrow Q \longrightarrow 1$$
with Euler class $\zeta \in H^2(Q;\mathbb{Z})$.  Let $Q'$ be a normal subgroup of $Q$ and let $\zeta' \in H^2(Q';\mathbb{Z})$ be the restriction of $\zeta$.  Finally, consider some $\zeta'' \in H^2(Q;\mathbb{Z})$ such that $n \cdot \zeta'' = \zeta'$.  Corresponding to $\zeta''$, there is a central extension
$$1 \longrightarrow \mathbb{Z} \longrightarrow G'' \longrightarrow Q' \longrightarrow 1$$
which fits into a commutative diagram
$$\begin{array}{ccccccccc}
1 & \longrightarrow & \mathbb{Z} & \longrightarrow & G'' & \longrightarrow & Q' & \longrightarrow & 1\\
  & & \downarrow \times n & & \downarrow & & \downarrow & & \\
1 & \longrightarrow & \mathbb{Z} & \longrightarrow & G & \longrightarrow & Q & \longrightarrow & 1
\end{array}$$
The group $G''$ is thus a subgroup of $G$.

Question : What assumptions can I place on $\zeta''$ which would ensure that $G''$ is a normal subgroup of $G$?