Might be worth considering Sigma lenses. They do most of their range with the option of Sony mount, so that would give you access to these for example, all of which are highly regarded by wildlife photographers: -
120-400mm f4.5-5.6 OS HSM
150-500mm f5-6.3 RF DG OS
50-500mm f4-6.3 EX DG
300mm f2.8 EX DG
500mm f4.5 EX DG
And, don't forget that increasing the camera's ISO setting from 100 to 200, or from 200 to 400, etc. etc., will give you the equivalent of an extra stop of light. (and 100 to 400 will give 2-stops extra etc.), so where lighting conditions are good, you don't necessarily need f2.8.
Regards
Mike.
EDIT - eeyore beat me to it. 