  Let $f$ be an endomorphism of the projective line. There is a natural
conjugation action on the space of such morphisms by elements of the projective
linear group. The group of automorphisms, or stabilizer group, of a given $f$
for this action is known to be a finite group. We determine explicit families
that parameterize all endomorphisms defined over $\bar{\mathbb{Q}}$ of degree
$3$ and degree $4$ that have a nontrivial automorphism, the
\textit{automorphism locus} of the moduli space of dynamical systems. We
analyze the geometry of these loci in the appropriate moduli space of dynamical
systems. Further, for each family of maps, we study the possible structures of
$\mathbb{Q}$-rational preperiodic points which occur under specialization.
