D. J. Simms in his book "Lie groups and quantum mechanics" (page 9) says that: any representation of $\sigma$ of $\tilde{G}$ (the simplgy connected covering group of the Lie group) in $U(H)$ such that $\sigma (K)\subset U(1)$ (in which $K$ is the kernel of covering map $p:\tilde{G}\to G$) will defines a unique projective representation $T$ of $G$ in $U(\hat{H})$ such that $T\circ p=\pi \circ \sigma$ (where $\pi :U(H)\to U(\hat{H})$ sends every unitary operator $U:H\to H$ to the corresponding symmetry $S_U :PH\to PH$).
I would really appreciate if somone could explain the reason of the claim above. Thank you in advance.