  In relation to Itzkowitz's problem, we show that a $\mathfrak c$-bounded
$P$-group is balanced if and only if it is functionally balanced. We prove that
for an arbitrary $P$-group, being functionally balanced is equivalent to being
strongly functionally balanced. A special focus is given to the uniform free
topological group defined over a uniform $P$-space. In particular, we show that
this group is (functionally) balanced precisely when its subsets $B_n,$
consisting of words of length at most $n,$ are all (resp., functionally)
balanced.
