Remind that a group $G$ has property FH if any $G$ action on a real Hilbert space has a fixed point. For $\sigma$-compact, locally compact groups this is equivalent to the celebrated Kazhdan's property (T).
Is property FH stable under extensions? i.e. if $$ is an extension of a group with property FH by a group with property FH, does it follow that $$ has property FH?
The similar statement concerning property (T) is true for locally compact groups, cf p. 69 of Kazhdan's Property (T) by Bekka, de la Harpe, Valette, but I am really interested in property FH.
I assume that by an "extension" of $N$ by $Q$ you mean a short exact sequence $$ 1\to N\to G\to Q\to 1. $$ Thus, $N$ can be regarded as a normal subgroup of $G$ and $G/N\cong Q$ (as a topological group).
Suppose that we have a continuous isometric affine action of $G$ on a Hilbert space $H$. By the FH property of $N$, there exists a nonempty closed affine subspace $A\subset H$, the fixed point set of $N$. Hence, $A$ is also isometric to a Hilbert space. Since $N$ is normal in $G$, the subspace $A$ is $G$-invariant. The group $Q$ then acts continuously and isometrically on $A$ (via the action of $G$, since $N$ acts trivially on $A$). Hence, by the property FH of $Q$, the group $Q$ has a fixed point $a\in A$. This point is fixed by $G$, of course.