On automorphism groups of half-arc-transitive tetravalent graphs

We characterize connected tetravalent graphs $\mathrm{\Gamma }$ which admit groups $M<H$ of automorphisms such that $\mathrm{\Gamma }$ is $M$-half-arc-transitive and $H$-arc-transitive. Examples for each case are constructed, including a counter-example to a question asked by A. R. Rivera and P. Šparl in 2019 as well as the first example of tetravalent normal-edge-transitive non-normal Cayley graph on a nonabelian simple group.