An example of this concept is to take the 2-sphere $S^2$ then one can calculate that $${\cal{M}}(S^2)=1,$$ but $${\cal{M}}^*(S^2)={\mathbb{Z}}_2.$$

For the genus one orientable surface, i.e. the torus $T=S^1\times S^1$ it is known that its (extended) mapping class group $${\cal{M}}^*(T)=GL_2({\mathbb{Z}}),$$ but usually by the (non-extended) mapping class group, that is, the group of isotopy classes of homeomorphisms that preserve orientations (the Dehn's twists) is just $${\cal{M}}(T)=SL_2({\mathbb{Z}}).$$

In these two examples we see that $\cal{M}^*$ is an extension of $\cal{M}$ by ${\mathbb{Z}}_2$ trivial for the 2-sphere and non trivial for the torus.

For the projective plane ${\mathbb{R}}P^2$ we have $${\cal{M}}({\mathbb{R}}P^2)={\cal{M}}^*({\mathbb{R}}P^2)=1$$

And what about the Klein bottle? $${\cal{M}}(K)={\mathbb{Z}}_2$$ $${\cal{M}}^*(K)={\mathbb{Z}}_2\oplus{\mathbb{Z}}_2$$