There does not exist a reasonable necessary and sufficient condition for an infinite centerless group to be complete. More precisely, letting $V$ be the set-theoretic universe, there exists an infinite complete group $G \in V$ and a $c.c.c$ notion of forcing $\mathbb{P}$ such that $G$ has an outer automorphism in the generic extension $V^{\mathbb{P}}$. An example can be found in :

S. Thomas, The automorphism tower problem II, Israel J. 
Math. 103 (1998), 93-109.

For more along these lines, see:

J. Hamkins and S. Thomas, 
Changing the heights of automorphism towers, Annals Pure Appl. Logic 102 (2000), 139-157.