Quantile regression: how to determine to which quantile an individual belongs

Dear Forum

I am running a quantile regression (for the 0.1, 0.5 and 0.9 quantile) on a cross-section data set containing individuals from the Danish labor market aged 40-55.
My analysis should try to detect whether there is an unexplained gender wage gap in my sample (a negative coefficient on my gender dummy). My LHS-variable is log(wage).
I divide my control variables into 2 groups (see below) and start by controlling for group 1. After having run my regression on the variables from group 1, I add group 2.
Group 1: experience, tenure and education.
Group 2: sector, industry, geography and other variables specific to the job.

After having controlled for group 2 variables, the gender wage gap seems to increase at the median compared to the gap estimated in the model that only includes group 1 variables.
Therefore I would like to find out why this is so by investigating whether women (at the median) primarily work in that industry, geographic region etc. that pays the highest wages.
In order to do this, I have considered obtaining the residuals for each individual from the model with both group 1 and 2 variables. Then I would find the quantiles of the residuals. By comparing the residual of each individual with the quantiles just defined, I determine which quantile each individual belongs to.
In that way I hope to be able to say what characterizes the women belonging to the median by looking at their specific jobs.

Do any of you know whether the residuals can be used in this way for defining the quantile each individual belongs to? References to work dealing with this issue would be highly appreciated.

I thank you in advance!

Best regards,