nt.number theory - Distinguishing congruence subgroups of the modular group

This question is something of a follow-up to
Transformation formulae for classical theta functions .

How does one recognise whether a subgroup of the modular group
$Gamma=mathrm{SL}_2(mathbb{Z})$ is a congruence subgroup?

Now that's too broad a question for me to expect a simple answer
so here's a more specific question. The subgroup $Gamma_1(4)$
of the modular group is free of rank $2$ and freely generated by
$A=left(
begin{array}{cc}
1&1\
0&1
end{array}right)$
and
$B=left(
begin{array}{cc}
1&0\
4&1
end{array}right)$. If $zeta$ and $eta$ are roots of unity there is a
homomorphism $phi$ from $Gamma_1(4)$ to the unit circle group
sending $A$ and $B$ to $zeta$ and $eta$ resepectively. Then the kernel $K$
of $phi$ has finite index in $Gamma_1(4)$. How do we determine whether $K$
is a congruence subgroup, and if so what its level is?

In this example, the answer is yes when $zeta^4=eta^4=1$. There are
also examples involving cube roots of unity, and involving eighth
roots of unity where the answer is yes. I am interested in this example
since one can construct a "modular function" $f$, homolomorphic on
the upper half-plane and meromorphic at cusps such that $f(Az)=phi(A)f(z)$
for all $AinGamma_1(4)$. One can take $f=theta_2^atheta_3^btheta_4^c$
for appropriate rationals $a$, $b$ and $c$.

Finally, a vaguer general question. Given a subgroup $H$ of
$Gamma$ specified as the kernel of a homomorphism from $Gamma$ or
$Gamma_1(4)$ (or something similar) to a reasonably tractable target group,
how does one determine whether $H$ is a congruence subgroup?