  We construct four infinite families of chiral $3$-polytopes of type $\{4,
8\}$, with $1024m^4$, $2048m^4$, $4096m^4$ and $8192m^4$ automorphisms for
every positive integer $m$, respectively. The automorphism groups of these
polytopes are solvable groups, and when $m$ is a power of $2$, they provide
examples with automorphism groups of order $2^n$ where $n \geq 10$. (On the
other hand, no chiral polytopes of type $\{4, 8\}$ exist for $n \leq 9$.) In
particular, our families give a partial answer to a problem proposed by Schulte
and Weiss in [Problems on polytopes, their groups, and realizations, {\em
Period. Math. Hungar.} 53 (2006), 231-255] and a problem proposed by Pellicer
in [Developments and open problems on chiral polytopes, {\em Ars Math. Contemp}
5 (2012), 333-354].
