To deal with isogenies it is useful to start with some of the basic work done on the Congruence Subgroup Problem in the 1960s and later. This problem can be formulated for any connected linear algebraic group defined over a number field such as $\mathbb{Q}$. Much of the work focuses on the difficult case of an (almost) simple group $G$, where it was known long ago for $G=\mathrm{SL}_2$ that not every arithmetic subgroup of $G(\mathbb{Q})$ is a congruence subgroup (a subgroup containing a principal congruence subgroup, necessarily of finite index in $G(\mathbb{Q})$). But eventually the split, simply connected groups of rank at least 2 (such as $\mathrm{SL}_n$ with $n \geq 3$) were found to satisfy the congruence subgroup property unless the field is totally imaginary (Bass-Milnor-Serre, Matsumoto). Further work by Prasad-Raghunathan has dealt with non-split groups.

But when $G$ fails to be simply connected the outcome is much less favorable. This is discussed carefully by Serre in the Bourbaki seminar (exp. 330) here. See 1.2(c); later work clarified the status of the strong approximation property invoked here.

In particular, the isogeny from the simply connected covering group $\widehat{G}$ onto $G$ does not behave at all well on congruence subgroups. If I understand correctly what is being asked here, this often provides a strongly negative answer.

To deal with isogenies it is useful to start with some of the basic work done on the Congruence Subgroup Problem in the 1960s and later. This problem can be formulated for any connected linear algebraic group defined over a number field such as $\mathbb{Q}$. Much of the work focuses on the difficult case of an (almost) simple group $G$, where it was known long ago for $G=\mathrm{SL}_2$ that not every arithmetic subgroup of $G(\mathbb{Q})$ (commensurate to $G(\mathbb{Z})$) is a congruence subgroup (a subgroup containing a principal congruence subgroup, necessarily of finite index in $G(\mathbb{Z})$). But eventually the split, simply connected groups of rank at least 2 (such as $\mathrm{SL}_n$ with $n \geq 3$) were found to satisfy the congruence subgroup property unless the field is totally imaginary (Bass-Milnor-Serre, Matsumoto). Further work by Prasad-Raghunathan has dealt with non-split groups.

But when $G$ fails to be simply connected the outcome is much less favorable. This is discussed carefully by Serre in the Bourbaki seminar (exp. 330) here. See 1.2(c); later work clarified the status of the strong approximation property invoked here.

In particular, the isogeny from the simply connected covering group $\widehat{G}$ onto $G$ does not behave at all well on congruence subgroups. If I understand correctly what is being asked here, this often provides a strongly negative answer.

To deal with isogenies it is useful to start with some of the basic work done on the Congruence Subgroup Problem in the 1960s and later. This problem can be formulated for any connected linear algebraic group defined over a number field such as $\mathbb{Q}$. Much of the work focuses on the difficult case of an (almost) simple group $G$, where it was known long ago that not every arithmetic subgroup of $\mathrm{SL}_2$ is a congruence subgroup (a subgroup containing a principal congruence subgroup, necessarily of finite index). But eventually the split, simply connected groups of rank at least 2 (such as special linear groups) were found to satisfy the congruence subgroup property unless the field is totally imaginary (Bass-Milnor-Serre, Matsumoto). Further work by Prasad-Raghunathan has dealt with non-split groups.

But when $G$ fails to be simply connected the outcome is much less favorable. This is discussed carefully by Serre in the Bourbaki seminar (exp. 330) here. See 1.2(c); later work clarified the status of the strong approximation property invoked here.

In particular, the isogeny from the simply connected covering group $\widehat{G}$ onto $G$ does not behave at all well on congruence subgroups. If I understand correctly what is being asked here, this often provides a strongly negative answer.

To deal with isogenies it is useful to start with some of the basic work done on the Congruence Subgroup Problem in the 1960s and later. This problem can be formulated for any connected linear algebraic group defined over a number field such as $\mathbb{Q}$. Much of the work focuses on the difficult case of an (almost) simple group $G$, where it was known long ago for $G=\mathrm{SL}_2$ that not every arithmetic subgroup of $G(\mathbb{Q})$ is a congruence subgroup (a subgroup containing a principal congruence subgroup, necessarily of finite index in $G(\mathbb{Q})$). But eventually the split, simply connected groups of rank at least 2 (such as $\mathrm{SL}_n$ with $n \geq 3$) were found to satisfy the congruence subgroup property unless the field is totally imaginary (Bass-Milnor-Serre, Matsumoto). Further work by Prasad-Raghunathan has dealt with non-split groups.

But when $G$ fails to be simply connected the outcome is much less favorable. This is discussed carefully by Serre in the Bourbaki seminar (exp. 330) here. See 1.2(c); later work clarified the status of the strong approximation property invoked here.

In particular, the isogeny from the simply connected covering group $\widehat{G}$ onto $G$ does not behave at all well on congruence subgroups. If I understand correctly what is being asked here, this often provides a strongly negative answer.