Hi. Avalon is a good feeder into the Grammar Schools, with pass rates approaching 100%.
Kingsmead is more of an all-through school but it finishes at 16 and so there is still a requirement to move on. Until recently it had boarders and has the reputation of being geared a little at those children who need a little extra help.
The other option is Birkenhead Prep, which feeds into Birkenhead School. This is the local HMC public school and unlike the Grammar Schools is co-ed. http://www.birkenheadschool.co.uk/welcome-prep