  We construct solutions to the heterotic G$_2$ system on almost contact metric
manifolds with reduced characteristic holonomy. We focus on
$3$-$(\alpha,\delta)$-Sasaki manifolds and $(\alpha,\delta)$-Sasaki manifolds,
the latter being a convenient reformulation of spin $\eta$-Einstein
$\alpha$-Sasaki manifolds. Investigating a $1$-parameter family of
G$_2$-connections on the tangent bundle, we obtain several approximate
solutions as well as one new class of exact solutions on degenerate
$3$-$(\alpha,\delta)$-Sasaki manifolds.
