The g-extra connectivity is a very important index to evaluate the fault tolerance, reliability of interconnection networks. Let g be a non-negative integer, G be a connected graph with vertex set V and edge set E, a subset $S\subseteq V$ is called a g-extra cut of G if the graph induced by the set $G-S$ is disconnected and each component of $G-S$ has at least $g+1$ vertices. The g-extra connectivity of G, denoted as ${\kappa }_g\left(G\right)$, is the cardinality of the minimum g-extra cut of G. Mycielski introduced a graph transformation to discover chromatic numbers of triangle-free graphs that can be arbitrarily large. This transformation converts a graph G into a new compound graph called $\mu \left(G\right)$, also known as the Mycielskian graph of G. In this paper, we study the relationship on g-extra connectivity between the Mycielskian graph $\mu \left(G\right)$ and the graph G. In addition, we show that ${\kappa }_3\left(\mu \left(G\right)\right)=2{\kappa }_1\left(G\right)+1$ for ${\kappa }_1\left(G\right)\le min\{4,\lfloor \frac{n}{2}\rfloor \}$, and prove the bounds of ${\kappa }_{2g+1}\left(\mu \left(G\right)\right)$ for $g\ge 2$.