New outer automorphism for $G_1 \times G_2$

Question

Suppose $G_1$ is a group, which has no outer automorphism.

Suppose $G_2$ is a group, which has no outer automorphism.

Main Question: What are the possible conditions to know can there be an outer automorphism for $$ G_1 \times G_2? $$

An easier question:

Suppose $G_1=SU(2)$ is a group, which has no outer automorphism.

Suppose $G_2= \mathbb{Z}/2\mathbb{Z}$ is a group, which has no outer automorphism.

• $SU(2) \times SU(2)$ has however a order-2 outer automorphism.
• $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ has however a order-2 outer automorphism.

Simpler Question: Is there an outer automorphism for $$ SU(2) \times \mathbb{Z}/2\mathbb{Z} ? $$

If you can answer anything of the above two Questions - this counts as a perfect answer.

Answer 1

A sufficient condition is that $G_1$ has a central subgroup $C$ isomorphic to a quotient of $G_2$ - this holds for your example.

We then have a natural map $q:G_2\to C$.

The map $\phi:G_1\times G_2\to G_1\times G_2:(g,h)\mapsto (gq(h),h)$ is an outer automorphism.