Hecke eigensystems of automorphic forms (mod p) of Hodge type and algebraic modular forms

We show that the systems of prime-to-$p$ Hecke eigenvalues arising from automorphic forms$\pmod p$ for a good prime $p$ associated to an algebraic group $G/\mathbb Q$ of Hodge type are the same as those arising from algebraic modular forms$\pmod p$ associated to an inner form of $G$. As an application we give an explicit upper bound for the number of those systems arising from automorphic forms$\pmod p$ associated to the similitude group $G$ of a Hermitian form with a totally indefinite quaternionic multiplication. We also show that the superspecial locus of the moduli space of abelian varieties of this type, possibly having bad reduction at $p$, is non-empty.