A generalisation of the pencil of Kuribayashi-Komiya quartics

The pencil of Kuribayashi-Komiya quartics $$x^4+y^4+z^4+t(x^2{y}^2+x^2{z}^2+y^2{z}^2)=0wheret\in \bar{\mathbb{C}}$$is a complex one-dimensional family of Riemann surfaces of genus three endowed with a group of automorphisms isomorphic to the symmetric group of order twenty-four. This pencil has been extensively studied from different points of view. This paper is aimed at studying, for each prime number $p⩾5$, the pencil of $\mathit{\text{generalised}}\mathit{\text{Kuribayashi-Komiya}}\mathit{\text{curves}}$ ${\mathcal{F}}_p$, given by the curves $$x^{2p}+y^{2p}+z^{2p}+t(x^p{y}^p+x^p{z}^p+y^p{z}^p)=0wheret\in \bar{\mathbb{C}}\mathrm{.}$$We determine the full automorphism group $G$ of each smooth member $X\in {\mathcal{F}}_p$ and study the action of $G$ and of its subgroups on $X$. In particular, we show that no member of the pencil is hyperelliptic. As a by-product, we derive a classification of all those Riemann surfaces of genus $(p-1)(2p-1)$ that are endowed with a group of automorphisms isomorphic to the full automorphism group of the generic smooth member of ${\mathcal{F}}_p\mathrm{.}$