No trace of this name or similar at Christchurch, Oxford at a suitable time; one at University College, which might perhaps fit, but unsure. If so:
Foster: son of Thomas of Carlby, Lincolnshire, sacerd. (CCEd 113278); UNIVERSITY COLLEGE, matric. 4 Jul. 1634 aged 17; one of these names V of Brixworth, Northants 1651 (etc.); his father compounded for Carlby 18 Mar. 1585/6 [while another Thomas instituted to that rectory 9 Apr. 1625 (CCEd 130722)]; see Foster Ind. Eccles.