How to design a finite state machine to detect 0110 in the input?
It looks like you're missing the overlap case, if I understand your diagram correctly.  This does the trick:

A    // goto B if read 0 else stay
B    // goto C if read 1 else stay
C    // goto D if read 1 else goto B
D    // goto E if read 0 else goto A
E    // emit rising edge, goto B


assuming that you have a falling edge after every read.