Fixed points of automorphisms of certain non-cyclic $p$-groups and the dihedral group

Let $G=\mathbf{Z}_{p} \oplus \mathbf{Z}_{p^2}$, where $p$ is a prime number. Suppose that $d$ is a divisor of the order of $G$. In this paper we find the number of automorphisms of $G$ fixing $d$ elements of $G$, and denote it by $\theta(G,d)$. As a consequence, we prove a conjecture of Checco-Darling-Longfield-Wisdom. We also find the exact number of fixed-point-free automorphisms of the group $\mathbf{Z}_{p^{a}} \oplus \mathbf{Z}_{p^{b}}$, where $a$ and $b$ are positive integers with $a<b$. Finally, we compute $\theta(D_{2q},d)$, where $D_{2q}$ is the dihedral group of order $2q$, $q$ is an odd prime and $d \in \{1,q,2q\}$.