Hurwitz's theorem on automorphisms tells us that the group of automorphisms of a nonsingular complex algebraic curve of genus at least 2 is bounded above by $84(g-1)$ where $g$ is the genus of the curve. We also know that for infinitely many $g$, this bound is sharp.

My question is whether these curves are 'isolated' or not. More precisely, given a fixed genus $g$, can I find a a one parameter family of curves which achieve the maximal bound for the automorphism group.

Of course there are some $g$ for which the $84(g-1)$ bound is not achieved, but I am happy with a family of curves that have the largest occurring automorphism group for that genus. Along these lines, I would also be interested in some "highly symmetric" (large automorphism group) one parameter family of curves of given genus.

Any insight/references/ways-to-think-about-this would be of great help!