First off, zero chance of Turkey letting Greece take over the Aegaeis.
Also, the Turkish Navy is rather busy in the Mediterranean, and the Eastern Mediterranean is one of the focus point of naval power today. There are good reasons for NATO to have SNMG1, SNMG2 and SNMCMG2, the UN having MTF448, and various navies having further forces permanently in the area.
I was actually wondering with my question whether Fleet Command has any units (in particular submarines) permanently stationed in the Black Sea, or whether it's only the support and patrol units of the Northern Sea Area Command, and of course the FAC squadron stationed in Istanbul, primarily responsible for Marmama but probably also with a control zone further north.
The Surface Action Group is also headquartered in Kocaeli on Marmara though.
I know most of the Navy is homeported at Aksaz.