  We study cyclically presented groups of type $\mathfrak{F}$ to determine when
they are perfect. It turns out that to do so, it is enough to consider the
Prishchepov groups, so modulo a certain conjecture, we classify the perfect
Prishchepov groups $P(r,n,k,s,q)$ in terms of the defining integer parameters
$r,n,k,s,q$. In particular, we obtain a classification of the perfect Campbell
and Robertson's Fibonacci-type groups $H(r,n,s)$, thereby proving a conjecture
of Williams, and yielding a complete classification of the groups $H(r,n,s)$
that are connected Labelled Oriented Graph groups.
