The object of size 84

$N_G(C_2)$ is a dihedral group of order 8, so the automorphism group is $C_2\times C_2$, abelian, and its identification with the automorphisms of every incarnation is canonical. $D_8$ has exactly three subgroups of order 4 containing its centre, one in each of the classes $C_4$, $V_4^a$ and $V_4^b$, so the three involutions have these quotient classes, and the quotient class names the involution. The power is not available, since an involution $t$ has $C_G(t)\neq\langle t\rangle$.

Reversing a Coxeter arc and pairing the four bitangents at a centre into Coxeter edges both have quotient class $C_4$, and the seam sending a centre $c$ with a bitangent $\ell$ to the Coxeter arc from the vertex of $\ell$ to its neighbour fixed by the involution with centre $c$ matches them. Replacing a Sylow 3-subgroup by the one generating $G$ with it is $C_4$, and by the one generating an $A_4$ of class $a$ or $b$ is $V_4^a$ or $V_4^b$; exchanging the ordered pair of vertices of a quadrangle is $V_4^a$; passing to the other pairing of four points of $\Proj^1(\F_7)$ with cross-ratio $\{2,4\}$ is $C_4$.