Do Riemann surfaces with large automorphism groups have long systoles?

Question

In the Wikipedia page on the Klein quartic, it says, "Whilst the Klein quartic maximises the symmetry group for surfaces of genus 3, it does not maximise the systole length." In the Wikipedia page on Bring's curve, it claims, "Similarly to the Klein quartic, Bring's surface does not maximize the systole length among compact Riemann surfaces in its topological category (that is, surfaces having the same genus) despite maximizing the size of the automorphism group." The Bolza surface has the largest automorphism group and the longest systole for surfaces of genus 2. The usage of the words whilst and despite makes me think that Riemann surfaces with a large automorphism group have a long systole. Is this generally true?