For which integers $n$ does every surjection $SL_2(\mathbb{Z})\twoheadrightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$?

(this is the usual kernel, ie, the subgroup of matrices congruent to 1 mod $n$).

A positive answer (for some $n$) would certainly imply that $\Gamma(n)$ is characteristic inside $SL_2(\mathbb{Z})$, which is true for even $n$ by David Speyer's beautiful answer (pointing to a result of Hua and Reiner) here:

are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?

From his answer, it's clear that our statement is false for odd $n$. However, even for $n\equiv 0\mod 2$, what's strange is that after doing some computations, the statement is true for $n = 2,4,6,\ldots, 24$, is false for $n = 26$, is true for $n = 28,30,32$, and is false again for $n = 34$.

Has there been any work on this? (I always find questions about $SL_2(\mathbb{Z})$ hard to google, due to the difficulty of making queries involving symbols).

EDIT: So far, after computing examples for even $n$ ranging from 2 through 58, the even $n$ for which the statement is false are: $$26,\ldots, 34,\ldots, 38,\ldots, 46,\ldots, 52, 54,\ldots, 58$$ where "$\ldots$" indicate gaps.

For which integers $n$ does every surjection $SL_2(\mathbb{Z})\twoheadrightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$?

(this is the usual kernel, ie, the subgroup of matrices congruent to 1 mod $n$).

A positive answer (for some $n$) would certainly imply that $\Gamma(n)$ is characteristic inside $SL_2(\mathbb{Z})$, which is true for even $n$ by David Speyer's beautiful answer (pointing to a result of Hua and Reiner) here:

are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?

From his answer, it's clear that our statement is false for odd $n$. However, even for $n\equiv 0\mod 2$, what's strange is that after doing some computations, the statement is true for $n = 2,4,6,\ldots, 24$, is false for $n = 26$, is true for $n = 28,30,32$, and is false again for $n = 34$.

Has there been any work on this? (I always find questions about $SL_2(\mathbb{Z})$ hard to google, due to the difficulty of making queries involving symbols).

EDIT: So far, after computing examples for even $n$ ranging from 2 through 58, the even $n$ for which the statement is false are: $$26,\ldots, 34,\ldots, 38,\ldots, 46,\ldots, 52, 54,\ldots, 58$$ where "$\ldots$" indicate gaps.

For which integers $n$ does every surjection $SL_2(\mathbb{Z})\twoheadrightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$?

(this is the usual kernel, ie, the subgroup of matrices congruent to 1 mod $n$).

A positive answer (for some $n$) would certainly imply that $\Gamma(n)$ is characteristic inside $SL_2(\mathbb{Z})$, which is true for even $n$ by David Speyer's beautiful answer (pointing to a result of Hua and Reiner) here:

are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?

From his answer, it's clear that our statement is false for odd $n$. However, even for $n\equiv 0\mod 2$, what's strange is that after doing some computations, the statement is true for $n = 2,4,6,\ldots, 24$, is false for $n = 26$, is true for $n = 28,30,32$, and is false again for $n = 34$.

Has there been any work on this? (I always find questions about $SL_2(\mathbb{Z})$ hard to google, due to the difficulty of making queries involving symbols).

For which integers $n$ does every surjection $SL_2(\mathbb{Z})\twoheadrightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$?

(this is the usual kernel, ie, the subgroup of matrices congruent to 1 mod $n$).

A positive answer (for some $n$) would certainly imply that $\Gamma(n)$ is characteristic inside $SL_2(\mathbb{Z})$, which is true for even $n$ by David Speyer's beautiful answer (pointing to a result of Hua and Reiner) here:

are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?

From his answer, it's clear that our statement is false for odd $n$. However, even for $n\equiv 0\mod 2$, what's strange is that after doing some computations, the statement is true for $n = 2,4,6,\ldots, 24$, is false for $n = 26$, is true for $n = 28,30,32$, and is false again for $n = 34$.

Has there been any work on this? (I always find questions about $SL_2(\mathbb{Z})$ hard to google, due to the difficulty of making queries involving symbols).

EDIT: So far, after computing examples for even $n$ ranging from 2 through 58, the even $n$ for which the statement is false are: $$26,\ldots, 34,\ldots, 38,\ldots, 46,\ldots, 52, 54,\ldots, 58$$ where "$\ldots$" indicate gaps.