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It is shown that if a finite generically smooth morphism   $f\,:\,Y\,\longrightarrow \, X$   of smooth projective varieties induces an isomorphism of the étale fundamental groups, then the induced map of the stratified fundamental groups   $\pi _{1}^{str}(f)\,:\, \pi _{1}^{str}(Y,\, y)\,\longrightarrow \, \pi _{1}^{str}(X,\, f(y))$   is also an isomorphism.

Isomorphism of Étale Fundamental Groups Lifts to Isomorphism of Stratified Fundamental Group