Automorphisms of strongly regular graphs with parameters $(1305,440,115,165)$

A graph $\varGamma$ is called $t$-isoregular if, for any $i\le t$ and any $i$-vertex subset $S$, the number $\varGamma(S)$ depends only on the isomorphism class of the subgraph induced by $S$. A graph $\varGamma$ on $v$ vertices is called absolutely isoregular if it is $(v-1)$-isoregular. It is known that each $5$-isoregular graph is absolutely isoregular, and such graphs have been fully described. Each exactly $4$-isoregular graph is either a pseudogeometric graph for pG$_r(2r,2r^3+3r^2-1)$ or its complement. By Izo$(r)$ we denote a pseudogeometric graph for pG$_r(2r,2r^3+3r^2-1)$. Graphs Izo$(r)$ do not exist for an infinite set of values of $r$ ($r=3,4,6,10,\ldots$). The existence of Izo$(5)$ is unknown. In this work we find possible automorphisms for the neighborhood of an edge from Izo$(5)$.