  In a previous work we have introduced the notion of embedded
$\mathbf{Q}$-resolution, which essentially consists in allowing the final
ambient space to contain abelian quotient singularities, and A'Campo's formula
was calculated in this setting. Here we study the semistable reduction
associated with an embedded $\mathbf{Q}$-resolution so as to compute the mixed
Hodge structure on the cohomology of the Milnor fiber in the isolated case
using a generalization of Steenbrink's spectral sequence. Examples of
Yomdin-L\^{e} surface singularities are presented as an application.
