Groups with some central automorphisms fixing the central kernel quotient

Let $G$ be a group. An automorphism $\alpha$ of a group $G$ is called a central automorphism, if $x^{-1}x^{\alpha}\in Z(G)$ for all $x\in G$. Let $L_c(G)$ be the central kernel of $G$, that is the set of elements of $G$ fixed by all central  automorphisms of $G$ and $Aut_{L_c}(G)$ denote the group of all central automorphisms of $G$ fixing $G/L_c(G)$ element-wise. In the present paper, we investigate the properties of such automorphisms. Moreover, a full classification of $p$-groups $G$ of order at most $p^5$ where $Aut_{L_c}(G)=Inn(G)$ is also given.