There is another type of important morphism between the (orbifold) fundamental groups of the moduli spaces $M_{g,\nu}\rightarrow M_{g',\nu'}$ that is considered in Grothendieck's tower. You can see this morphism in three different ways. One is directly on the surfaces of type $(g,\nu)$ and $(g',\nu')$ (of genus $g$ with $\nu$ boundary components, resp. genus $g'$ with $\nu'$ boundary components). This morphism exists if you can put a set of disjoint simple closed loops on the surface of type $(g',\nu')$ such that when you cut along them, you cut your surface into one piece of type $(g,\nu)$, or else into several pieces of which at least one is of type $(g,\nu)$. You can also think of including the smaller surface of type $(g,\nu)$ into the bigger one by gluing it to other smaller pieces along the edges of their boundary components, to form the bigger one of type $(g',\nu')$ (which is the image Grothendieck had in mind when he talked about Lego).

The second way to see this morphism is as a morphism of moduli spaces, where $M_{g,\nu}$ is mapped to a boundary component of the Deligne-Mumford compactification $\overline{M}_{g',\nu}$, in fact precisely the boundary component corresponding to taking the simple closed loops on the surface of type $(g',\nu')$ that "cut out" the one of type $(g,\nu)$ and shrinking them to length zero, so they become nodes.

The third way to view this same morphism is on the fundamental groups. This is pretty easy, since the (orbifold) fundamental group of $M_{g,\nu}$ is generated by Dehn twists along simple closed loops on the surface of type $(g,\nu)$, and these just map to the Dehn twists along the same simple closed loops when the $(g,\nu)$ surface is included in the $(g',\nu')$ one as above.

The Teichmüller tower can be considered to be the collection of all the fundamental groups of the $M_{g,\nu}$ linked by the point-erasing morphisms and by these. Or, as Grothendieck wanted, instead of fundamental groups, that depends on a certain choice of base point, you can replace the groups by more symmetric fundamental groupoids based at all "tangential base points" on the moduli spaces.

The automorphism group of the Teichmüller tower basically then consists of tuples $(\phi_{g,\nu})$ such that each $\phi_{g,\nu}$ is an automorphism of $\pi_1(M_{g,\nu})$ and the different $\phi_{g,\nu}$ in the same tuple commute with the homomorphisms of the tower.

There is another type of important morphism between the (orbifold) fundamental groups of the moduli spaces $M_{g,\nu}\rightarrow M_{g',\nu'}$ that is considered in Grothendieck's tower. You can see this morphism in three different ways. One is directly on the surfaces of type $(g,\nu)$ and $(g',\nu')$ (of genus $g$ with $\nu$ boundary components, resp. genus $g'$ with $\nu'$ boundary components). This morphism exists if you can put a set of disjoint simple closed loops on the surface of type $(g',\nu')$ such that when you cut along them, you cut your surface into one piece of type $(g,\nu)$, or else into several pieces of which at least one is of type $(g,\nu)$. You can also think of including the smaller surface of type $(g,\nu)$ into the bigger one by gluing it to other smaller pieces along the edges of their boundary components, to form the bigger one of type $(g',\nu')$ (which is the image Grothendieck had in mind when he talked about Lego).

The second way to see this morphism is as a morphism of moduli spaces, where $M_{g,\nu}$ is mapped to a boundary component of the Deligne-Mumford compactification $\overline{M}_{g',\nu}$, in fact precisely the boundary component corresponding to taking the simple closed loops on the surface of type $(g',\nu')$ that "cut out" the one of type $(g,\nu)$ and shrinking them to length zero, so they become nodes.

The third way to view this same morphism is on the fundamental groups. This is pretty easy, since the (orbifold) fundamental group of $M_{g,\nu}$ is generated by Dehn twists along simple closed loops on the surface of type $(g,\nu)$, and these just map to the Dehn twists along the same simple closed loops when the $(g,\nu)$ surface is included in the $(g',\nu')$ one as above.

The Teichmüller tower can be considered to be the collection of all the fundamental groups of the $M_{g,\nu}$ linked by the point-erasing morphisms and by these. Or, as Grothendieck wanted, instead of fundamental groups, that depend on a certain choice of base point, you can replace the groups by more symmetric fundamental groupoids based at all "tangential base points" on the moduli spaces".

The automorphism group of the Teichmüller tower basically then consists of tuples $(\phi_{g,\nu})$ such that each $\phi_{g,\nu}$ is an automorphism of $\pi_1(M_{g,\nu})$ and the different $\phi_{g,\nu}$ in the same tuple commute with the homomorphisms of the tower.