TITLE: Interpolating between Krull-Schmidt and Jordan-Holder
QUESTION [7 upvotes]: Define a group $G$ to be semidirect-simple if it cannot be written as a semidirect product in a nontrivial way. (This is not a standard term.)
Suppose that
$$G_1 \ltimes (G_2 \ltimes (G_3 \ltimes ( \cdots \ltimes G_r ) \cdots ) ) \cong H_1 \ltimes (H_2 \ltimes (H_3 \ltimes ( \cdots \ltimes H_s ) \cdots ) )$$
with $G_1$, $G_2$, ..., $G_r$, $H_1$, $H_2$, ..., $H_s$ semidirect-simple. (Of course, the notation $\ltimes$ implicitly includes choices of various actions.)
Is it true that $r=s$ and $(G_1, G_2, \ldots, G_r)$ is a permutation of $(H_1, H_2, \ldots, H_s)$?
As mentioned in a previous question, I'm brainstorming group theory problems and tossing up the ones I don't know the answers to.

REPLY [5 votes]: Consider the dihedral group $D_8 = \langle x, y \rangle$ where $x$ has order $2$, $y$ has order $4$ and $xyx^{-1} = y^{-1}$. Then $D_8 = \langle x \rangle \ltimes \langle y \rangle$ and $D_8 = \langle xy \rangle \ltimes (\langle x \rangle \times \langle y^2 \rangle)$. That is,
$$D_8 \cong C_2 \ltimes C_4 \cong C_2 \ltimes (C_2 \ltimes C_2)$$
So the statement is not true.