Let $\mathbf{PG}(2,q^2)$ be the finite projective plane defined over the finite field $\mathbb{F}_{q^2}$. Then for each quadrangle, there is precisely one involution fixing it pointwise, and hence there is also precisely one involution fixing the subplane the quadrangle generates pointwise. This subplane is isomorphic to $\mathbf{PG}(2,q)$, and the involution is called a Baer involution.
But what about nonclassical finite projective planes ?
In other words: suppose that $\Gamma$ is an arbitrary axiomatic finite projective plane (of order $N$), which admits a Baer involution $\sigma$, fixing the subplane $\Gamma_\sigma$ (of order $\sqrt{N}$) elementwise.
Can there be other Baer involutions $\alpha \ne \sigma$ fixing the same plane $\Gamma_\sigma = \Gamma_\alpha$ elementwise ?
And if so, can they commute ?
More generally, what can we say about the group $H$ generated by all Baer involutions that fix the same subplane elementwise ?
