I am trying to understand Galois groups for a tower of fields

Please let me know if these examples are of any use. I am just employing some small groups.

You can construct a Galois extension $M/K$ whose Galois group is $S_3$, and then if $L$ is the intermediate field corresponding to $A_3 \cong C_3$ (I am writing $C_n$ for a cyclic group of order $n$), in your sequence you will have that $$ \operatorname{Gal}(L/K) \cong C_2, \qquad \operatorname{Gal}(M/L) \cong C_3. $$ But you can also construct a Galois extension $M/K$ whose Galois group is $C_6$, and then if $L$ is the intermediate field corresponding to $C_3$, in your sequence you will also have that $$ \operatorname{Gal}(L/K) \cong C_2, \qquad \operatorname{Gal}(M/L) \cong C_3. $$ Both exact sequences split, but you have different actions, and different groups $\operatorname{Gal}(M/K)$.

You can do the same for the groups $C_4$ and $V = C_2 \times C_2$. Taking for $L$ a subfield corresponding to a subgroup of order $2$, we have in both cases $$\operatorname{Gal}(L/K) \cong C_2 \cong \operatorname{Gal}(M/L),$$ but $V$ is a split extension, while $C_4$ is not.

Using the dihedral group of order $8$ and the quaternion group of order $8$ one can construct two towers such that $$\operatorname{Gal}(L/K) \cong V \qquad \operatorname{Gal}(M/L) \cong C_2,$$ here both sequences will be nonsplit, but the two groups $\operatorname{Gal}(M/K)$ are non-isomorphic.

Something seems to have gone wrong with the first claim. What is true is the following. If $\overline{\sigma}_1$ and $\overline{\sigma}_2$ are two extensions of $\sigma\in \operatorname{Gal}(L/K)$, then $\overline{\sigma}_2^{-1}\overline{\sigma}_1$ restricted to $L$ is $\sigma^{-1}\sigma$, i.e. the identity, so it is an element of $\operatorname{Gal}(M/L)$. In other words: $$ \overline{\sigma}_1=\overline{\sigma}_2 g $$ for some $g\in \operatorname{Gal}(M/L)$.

Note that the two lifts cannot always be conjugate to each other in the prescribed way. After all, for all we know $\operatorname{Gal}(M/K)$ might be abelian, and hence no two distinct elements in there would be conjugate in $\operatorname{Gal}(M/K)$.

[Edit] I cannot say anything about the other claim. At least not yet [/Edit]

In the abstract, the question you ask is known as the group extension problem. The wikipedia page should give you a good starting point for further reading.