Why must superpartners have the same gauge quantum numbers? The title leaves it quite clear, why must superpartners have the same gauge quantum numbers?
 A: A bosonic symmetry that acts differently on the different components of a supermultiplet is an R-symmetry. Such a symmetry does not commute with the supercharges. Since the commutator between an R-symmetry and a supercharge gives something Grassmann odd, it has to given another supercharge. Schematically
$$
[R,Q] = Q
$$
or in terms of variations acting on some field
$$
[ \delta_R , \delta_Q ] = \delta_Q
$$
If we gauge an R-symmetry then the variation $\delta_R$ is allowed to depend on the coordinates $x$. But then from the algebra above so is $\delta_Q$
$$
[ \delta_R(x) , \delta_Q(x) ] = \delta_Q(x)
$$
So gauging an R-symmetry leads to local supersymmetry. To see where this takes us we can consider the supersymmetry algebra
$$
\{ Q , Q \} = P
$$
or
$$
\{ \delta_Q , \delta_Q \} = \delta_P
$$
where $\delta_P$ is a translation. If $\delta_Q$ depends on $x$ we get
$$
\{ \delta_Q(x) , \delta_Q(x) \} = \delta_P(x)
$$
so local supersymmetry automatically leads to invariance under local translations or in other words under diffeomorphisms. Hence, the only way to gauge an R-symmetry is in supergravity. In particular in a gauged supergravity the gravitino is charged with respect to some gauge field while the graviton is not, so this gives an example where superpartners don't carry the same gauge quantum numbers.
