examples of outer automorphism group

It is easy to understand that $\mathrm{Out}\mathbb{Z}=\mathrm{Aut}\mathbb{Z}=\mathbb{Z}/2\mathbb{Z}$, since $\mathbb{Z}$ is abelian and there are no inner-automorphisms, save the trivial one.

Also, it is known that $\mathrm{Out}SL(2,\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}$

Another example is that, at least for orientable surfaces, the extended mapping class group (or the zeroth homeotopy group) of a surface $F$ is related to its fundamental group via ${ℳ}^{*}(F)=\mathrm{Out}({\pi }_1(F))$.

References

1. L. K. Hua, I Reiner Automorphisms of unimodular group, Trans Amer. Math. Soc. 71 (1951), 331-348.

2. H. Zieschang, E. Vogt, H. D. Coldewey, Surfaces and planar discontinuous groups, L.N.M. 875 (1981) Springer-Verlag.