  In this paper we study various rigidity aspects of the von Neumann algebra
$L(\Gamma)$ where $\Gamma$ is a graph product group \cite{Gr90} whose
underlying graph is a certain cycle of cliques and the vertex groups are the
wreath-like product property (T) groups introduced recently in \cite{CIOS21}.
Using an approach that combines methods from Popa's deformation/rigidity theory
with new techniques pertaining to graph product algebras, we describe all
symmetries of these von Neumann algebras and reduced C$^*$-algebras by
establishing formulas in the spirit of Genevois and Martin's results on
automorphisms of graph product groups \cite{GM19}.
