  We consider the sequence ${J_m,m \ge 2}$ of the 3-state Mealy automata over
an m-symbol alphabet such that the growth function of $J_m$ has the
intermediate growth order $[n ^{{\log n}/{2 \log m}} ]$. For each automaton
$J_m$ we describe the automaton transformation monoid $S_{J_m}$, defined by it,
provide generating series for the growth functions, and consider primary
properties of $S_{J_m}$ and $J_m$.
