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In the present paper, we classify groups with the same order and degree pattern as an almost simple group related to the projective special linear simple group $L_2\left(49\right)$. As a consequence of this result we can give a positive answer to a conjecture of W. J. Shi and J. X. Bi, for all almost simple groups related to $L_2\left(49\right)$ except $L_2\left(49\right)·2^2$. Also, we prove that if $M$ is an almost simple group related to $L_2\left(49\right)$ except $L_2\left(49\right)·2^2$ and $G$ is a finite group such that $\left|G\right|=\left|M\right|$ and $\Gamma \left(G\right)=\Gamma \left(M\right)$, then $G\cong M$.