Is she as smart as you are? Does her job require similar levels of knowledge and skill to yours?

(One, possible) short answer seems to be that when women move into male-dominated sectors, the pay falls.

"Occupations with a greater share of females pay less than those with a lower share, controlling for education and skill. This association is explained by two dominant views: devaluation and queuing. The [former] argues that the proportion of females in an occupation affects pay, owing to devaluation of work done by women. The [latter] views the pay offered in an occupation to affect its female proportion, due to employers’ preference for men–a gendered labor queue. Only a few past studies used longitudinal data, which is needed to test the theories. We use fixed-effects models, thus controlling for stable characteristics of occupations, and U.S. Census data from 1950 through 2000. We find substantial evidence for the devaluation view, but only scant evidence for the queuing view. ...

"The devaluation view does, however, posit a type of employer discrimination that occurs after jobs have a particular sex composition. The claim is that decisions of employers about the relative pay of “male” and “female” occupations are affected by gender bias. Employers ascribe a lower value for the work done in occupations with a high share of females and consequently set lower wage levels. ...

"To be sure, there is substantial inertia in both wage rates and sex composition as seen by the strong inter-period correlations (tables 1-3). Yet, using a longer time frame and decade-long lag, we have found substantial support for the view that increased feminization of oc-cupations diminishes their relative pay."

"Occupational Feminization and Pay: Assessing Causal Dynamics Using 1950-2000 U.S. Census Data " (https://www.researchgate.net/profile/Paul-Allison-4/publicat...)

"...the negative association between female percentage in occupations and their pay levels declines over time. However, when intervening variables are controlled for, the trend is reversed; the negative net effect of female percentage on occupational pay intensifies over time. This is true in the static analysis, when levels of education and work experience are controlled for at the individual and occupational levels, and also in the dynamic analysis, which uses a lagged dependent variable model to further reduce the risk of omitted variable bias. These two opposite processes [the entry of women into occupations requiring higher education, and the growing return to education and to occupations with higher educational requirements] reflect the upward occupational mobility of women, on the one hand, and its gendered consequences, on the other hand."

"A Second Look at the Process of Occupational Feminization and Pay Reduction in Occupations" (https://people.socsci.tau.ac.il/mu/hadasm/files/2019/03/Mand...)

Bonus:

"Females earn less than males. The reasons are not understood. This paper explores the theory that women ‘don’t ask’ and that they fail to ask out of concern for their personal relationships at work. Such an account is hard to test with standard labor-economics data sets. The paper therefore uses matched employer-employee data in which workers are directly questioned about their asking behaviour. It is unable to reject the hypotheses that males and females ask equally often for promotions and raises, and that men and women are equally concerned about relationships. Instead, the analysis documents evidence that women ‘don’t get’."

"DO WOMEN ASK?" (http://www.amandagoodall.com/artz29Dec2016ArtzetalGenderWome...)