Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$.

By congruence subgroup of $G_i$, for $i=1,2$, we means $K\cap G_i(\mathbb{Q})$, with $K$ compact open subgroup of $G_i(\mathbb{A}_f)$.

Then consider the images of congruence subgroups of $G_1$ under the map $f$, is it cofinal with the congruence of $G_2$, i.e. every congruence subgroup of $G_2$, it conatines image of some congruence of $G_1$ and every image of congruence subgroup of $G_1$ containes congruence subgroup of $G_2$ ?

If the above is not true, then is it true that almost all (except finite many) congruence subgroups of $G_2$ are contained in the image of some congruence subgroup of $G_1$?

For example $G_1=SL_2(\mathbb{Q})\times SL_2(\mathbb{Q})\times SL_2(\mathbb{Q})$, and $G_1$ has a natural map into $SP_8(\mathbb{Q})$ by tensor product action. Let $G_2$ be the image . Let $C_2$ (resp. $C_1$) be a connected shimura curve using $G_2$ (resp.$G_1$) and congruence subgroup $\Gamma_2$ (resp.$\Gamma_1$) of $G_2$ (resp. $G_1$). Then I want to know is it true that except finite many congruence subgroup $\Gamma_2$ , one can always find some $\Gamma_1$, such that there is an etale map from $C_2$ to $C_1$.