Homogeneous edge-disjoint $K_{2s}$ and $T_{st,t}$ unions

Let $r>2$ and $\sigma \in (0,r-1)$ be integers. We require $t<2s$, where $t=2^{\sigma +1}-1$ and $s=2^{r-\sigma -1}$. Generalizing a known $\{K_4,T_{6,3}\}$-ultrahomogenous graph $G_3^1$, we find that a finite, connected, undirected, arc-transitive graph $G_r^{\sigma }$ exists each of whose edges is shared by just two maximal subgraphs, namely a clique $X_0=K_{2s}$ and a $t$-partite regular-Tur'an graph $X_1=T_{st,t}$ on $s$ vertices per part. Each copy $Y$ of $X_i$ ($i=0,1$) in $G_r^{\sigma }$ shares each edge with just one copy of $X_{1-i}$ and all such copies of $X_{1-i}$ are pairwise distinct. Moreover, $G_r^{\sigma }$ is an edge-disjoint union of copies of $X_i$, for $i=0,1$. We prove that $G_r^{\sigma }$ is $\{K_{2s},T_{st,t}\}$-homogeneous if $t<2s$, and just $\{T_{st,t}\}$-homogeneous otherwise, meaning that there is an automorphism of $G_r^{\sigma }$ between any two such copies of $X_i$ relating two preselected arcs.