The Austin PD chief has come under intense scrutiny for his department's handling of various arrests and protests over the past few months, and after roughly half of the city council called on him to resign, it was reported that neither the city council or the mayor could legally fire him.[0]
The city manager, which is appointed and not elected, has the power to demote the police chief, but the state does not permit a police chief to be terminated.
Long story short, none of Austin's leaders have the power to fire the police chief.
0: https://www.texastribune.org/2020/06/18/austin-police-chief-...
edit: added reporting