Let $G$ be a finite non-abelian simple group, and let $\tilde{G}$ be a cover of $G$. What can we say about ${\rm Out}(\tilde{G})$ in relation to ${\rm Out}(G)$?
It is easy to show the natural map ${\rm Aut}(\tilde{G}) \to {\rm Aut}(G)$ induces an injection ${\rm Out}(\tilde{G}) \to {\rm Out}(G)$, as follows. First, the map ${\rm Aut}(\tilde{G}) \to {\rm Aut}(G)$ is injective: any element $\phi$ in the kernel induces a map $\tilde{G} \to Z(\tilde{G})$ (defined by $g \mapsto \phi(g) g^{-1}$) which must be trivial since $\tilde{G}$ is perfect. Thus any automorphism of $G$ lifts to at most one automorphism of $\tilde{G}$. Since any inner automorphism of $G$ lifts to an inner automorphism of $\tilde{G}$, it follows that the induced map ${\rm Out}(\tilde{G}) \to {\rm Out}(G)$ is an injection.
Are there any other general results one can say about ${\rm Out}(\tilde{G})$ in terms of ${\rm Out}(G)$? (Without invoking the classification theorem to enumerate all possibilities.) For example, can one use the value of $[\tilde{G}:G]$ to narrow down the possibilities even further?