  A crossed product functor is said to be injective if it takes injective
morphisms to injective morphisms. In this paper we show that every locally
compact group $G$ admits a maximal injective crossed product $A\mapsto
A\rtimes_{\inj}G$. Moreover, we give an explicit construction of this functor
that depends only on the maximal crossed product and the existence of
$G$-injective $C^*$-algebras; this is a sort of a `dual' result to the
construction of the minimal exact crossed product functor, the latter having
been studied for its relationship to the Baum-Connes conjecture. It turns out
that $\rtimes_\inj$ has interesting connections to exactness, the local lifting
property, amenable traces, and the weak expectation property.
