  Let $\mathbb{V}$ be a motivic variation of Hodge structure on a $K$-variety
$S$, let $\mathcal{H}$ be the associated $K$-algebraic Hodge bundle, and let
$\sigma \in \textrm{Aut}(\mathbb{C}/K)$ be an automorphism. The absolute Hodge
conjecture predicts that given a Hodge vector $v \in \mathcal{H}_{\mathbb{C},
s}$ above $s \in S(\mathbb{C})$ which lies inside $\mathbb{V}_{s}$, the
conjugate vector $v_{\sigma} \in \mathcal{H}_{\mathbb{C}, s_{\sigma}}$ is Hodge
and lies inside $\mathbb{V}_{s_{\sigma}}$. We study this problem in the
situation where we have an algebraic subvariety $Z \subset S_{\mathbb{C}}$
containing $s$ whose algebraic monodromy group $\mathbf{H}_Z$ fixes $v$. Using
relationships between $\mathbf{H}_Z$ and $\mathbf{H}_{Z_{\sigma}}$ coming from
the theories of complex and $\ell$-adic local systems, we establish a criterion
that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic
condition on $\mathbf{H}_{Z}$. We then use our criterion to establish new cases
of the absolute Hodge conjecture.
