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Let $f:X\to Y$ be a stratified map between Whitney stratified spaces such that for each stratum $S$ of $Y$, $f:f^{-1}(S)\to S$ is a proper stratified submersion. Let $\mathscr{T}_Y$ be a Thom–Mather control data of $Y$. Is there a Thom–Mather control data $\mathscr{T}_X$ on $X$ which is compatible with $\mathscr{T}_Y$ and $f$?

P.S. If $Y$ has only one stratum, this is a theorem of Mather which is used to prove Thom's first isotropy lemma.

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This is just a partial answer that might be useful. Your question is true if the map is slightly nicer, i.e., it is has the $a_f$ property (aka Thom property).

This is Theorem II.2.6 of the book of C. G. Gibson, K.Wirthmuller, A.A. du Plessis, E. J. N. Looijenga Topological Stability of Smooth Mappings

Given that not all maps admit a Thom stratification, see for example Sabbah's Morphismes analytiques stratifiés sans éclatement et cycles évanescents, your question is still uncertain.

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