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The locally compact abelian groups $G$ where ${\rm Aut}(G)$ has at most 2 topological orbits on $G$ are described in Theorem 2.13 of Markus Stroppel, Locally compact groups with few orbits under automorphisms. Topology Proc. 26 (2001/02), no. 2, 819–842.
You are asking for a table of $P(T)$ values for sporadic $T$. For non-sporadic simple groups $T$ a good reference is Table 4 of a 2015 paper by Guest-Morris-Praeger-Spiga. Warning: there is a typo for the value of $P(E_7)$, the factor $q^5-1$ should be $q^5+1$ (see Vassilev original paper).
@THC Your extra conditions seems to be asking whether Payne's Conjecture is true. This is a very hard open problem. See the paper by Bamberg, Glasby, Swartz, AS-configurations and skew-translation generalised quadrangles, J. Algebra 421 (2015), 311--330. This paper has some group theoretic constraints on $G$ and proves that there is no example with t=8 (using a computer search).
