Isometry groups of CAT(0) cube complexes

Given a CAT(0) cube complex X, we show that if Aut(X) $\mathrm{\ne }$ Isom(X) then there exists a full subcomplex of X which decomposes as a product with ${\mathbb{R}}^n$. As applications, we prove that if X is $\delta$-hyperbolic, cocompact and 1-ended, then Aut(X) $=$ Isom(X) unless X is quasi-isometric to ${\mathbb{H}}^2$, and extend the rank-rigidity result of Caprace-Sageev to any lattice $\mathrm{\Gamma }\le$ Isom(X).