$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is every normal subgroup of $\SL_2(\mathbb{Z}/n)$ also normal inside $\GL_2(\mathbb{Z}/n)$?

Of course, it suffices to ask the question when $n = p^m$ a prime power. In Classification of the Normal Congruence Subgroups of the Modular Group, McQuillan describes the normal subgroups of $\SL_2(\mathbb{Z}/p^m)$. When $p >3$, the only nontrivial normal subgroup is $\{\pm I\}$ (together with the kernels of maps to smaller prime powers). When $p = 3$, there is one additional possibility, of index 3, which one can check to be normal in $\GL_2(\mathbb{Z}/3^m)$.

When $p = 2$, on page 292 he describes the various normal subgroups of $\SL_2(\mathbb{Z}/2^m)$. However, amongst these possibilities, he writes that for $m\ge 4$, the matrix: $$A := \begin{bmatrix}1+2^{m-2} & 2^{m-1} \\ 2^{m-1} & 1-2^{m-1}\end{bmatrix}$$ generates a normal subgroup of $\SL_2(\mathbb{Z}/2^m)$, which seems to be incorrect as checked in GAP...which makes me a bit wary of his results.

In any case, for my purposes, I don't need a full classification, and I suspect my question probably has a simpler answer anyway.

EDIT: Actually I suppose a much weaker question is relevant for my purposes - Is it true that every normal subgroup of $\SL_2(\mathbb{Z}/n)$ is the intersection of a normal subgroup of $\GL_2(\mathbb{Z}/n)$ with $\SL_2(\mathbb{Z}/n)$? (As YCor points out in the comments, this is trivially equivalent to the original question. Living up to my name...)