We prove the existence and give constructions of a
$(p(k)-1)$
-fold perfect resolvable
$(v,k,1)$
-Mendelsohn design for any integers
$v>k\geq 2$
with
$v\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}\,k$
such that there exists a finite Frobenius group whose kernel
$K$
has order
$v$
and whose complement contains an element
$\unicode[STIX]{x1D719}$
of order
$k$
, where
$p(k)$
is the least prime factor of
$k$
. Such a design admits
$K\rtimes \langle \unicode[STIX]{x1D719}\rangle$
as a group of automorphisms and is perfect when
$k$
is a prime. As an application we prove that for any integer
$v=p_{1}^{e_{1}}\cdots p_{t}^{e_{t}}\geq 3$
in prime factorisation and any prime
$k$
dividing
$p_{i}^{e_{i}}-1$
for
$1\leq i\leq t$
, there exists a resolvable perfect
$(v,k,1)$
-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if
$k$
is even and divides
$p_{i}-1$
for
$1\leq i\leq t$
, then there are at least
$\unicode[STIX]{x1D711}(k)^{t}$
resolvable
$(v,k,1)$
-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where
$\unicode[STIX]{x1D711}$
is Euler’s totient function.