  We consider homotopy actions of a Lie algebroid on a graded manifold, defined
as suitable $L_{\infty}$-algebra morphisms. On the "semi-direct product" we
construct a homological vector field that projects to the Lie algebroid. Our
main theorem states that this construction is a bijection.
  Since several classical geometric structures can be described by homological
vector fields as above, we can display many explicit examples, involving Lie
algebroids (including extensions, representations up to homotopy and their
cocycles) as well as transitive Courant algebroids.
