  We show that for a locally compact group $G$, amongst a class which contains
amenable and small invariant neighbourhood groups, that its Fourier algebra
$A(G)$ satisfies a completely bounded version Pisier's similarity property with
similarity degree at most $2$. Specifically, any completely bounded
homomorphism $\pi: A(G)\to B(H)$ admits an invertible $S$ in $B(H)$ for which
$\|S\|\|S^{-1}\|\leq ||\pi||_{cb}^2$ and $S^{-1}\pi(\cdot)S$ extends to a
$*$-representation of the $C^*$-algebra $C_0(G)$. This significantly improves
some results due to Brannan and Samei (J. Funct. Anal. 259, 2010) and Brannan,
Daws and Samei (M\"{u}nster J. Math 6, 2013). We also note that $A(G)$ has
completely bounded similarity degree $1$ if and only if it is completely
isomorphic to an operator algebra if and only if $G$ is finite.
