Isomorphism of étale fundamental groups lifts to isomorphism of stratified fundamental group

It is shown that if a finite generically smooth morphism $f:Y⟶X$ of smooth projective varieties induces an isomorphism of the étale fundamental groups, then the induced map of the stratified fundamental groups ${\pi }_1^{str}(f):{\pi }_1^{str}(Y,y)⟶{\pi }_1^{str}(X,f(y))$ is also an isomorphism.