The two major GA players at Birmingham are Eurojet (operating 1 C500, 2 C550B & 2 C560XL) & Jet Options (operating 1 C550 & 1 C560). I'm not sure if they do air taxi work though.
Only other based GA is the Police chopper & a based Pa31 Panther.
Rumours around that HLX may look at another route into Birmingham in addition to CGN (STR perhaps?)
Fried Chicken