  Let $q$ be an odd prime, $m$ a positive integer, and let $\Ga_m(q)$ be the
group generated by two elements $x$ and $y$ subject to the relations
$x^{2m}=y^{qm}=1$ and $x^2=y^q$; that is, $\Ga_m(q)$ is the free product of two
cyclic groups of orders $2m$ respectively $qm$, amalgamated along their
subgroups of order $m$. Our main result determines the parity behaviour of the
generalized subgroup numbers of $\Ga_m(q)$ which were defined in [T. W.
M\"uller, Adv. in Math. 153 (2000), 118-154], and which count all the
homomorphisms of index $n$ subgroups of $\Ga_m(q)$ into a given finite group
$H$, in the case when $\gcd(m,| H|)=1$. This computation depends upon the
solution of three counting problems in the Hecke group $\mathfrak
H(q)=C_2*C_q$: (i) determination of the parity of the subgroup numbers of
$\mathfrak H(q)$; (ii) determination of the parity of the number of index $n$
subgroups of $\mathfrak H(q)$ which are isomorphic to a free product of copies
of $C_2$ and of $C_\infty$; (iii) determination of the parity of the number of
index $n$ subgroups in $\mathfrak H(q)$ which are isomorphic to a free product
of copies of $C_q$. The first problem has already been solved in [T. W.
M\"uller, in: {\it Groups: Topological, Combinatorial and Arithmetic Aspects},
(T. W. M\"uller ed.), LMS Lecture Notes Series 311, Cambridge University Press,
Cambridge, 2004, pp. 327-374]. The bulk of our paper deals with the solution of
Problems (ii) and (iii).
