comparing "pooled" coefficients between different models

Dear all,

I am running a mediation model with 6 continuous predictors and 2 mediators, separately for boys and girls. I want to test if the regression coefficients differ between gender models. For that, I have found the equation b1-b2/SQRT(s.eb1^2+ s.e.b2^2) and this works fine.

However, what I really want to do is extend this to test whether the coefficients of "clusters" of predictors differ - i.e. instead of testing each parameter one for one between the two models, I would like to test if together, both the predictors related to parenting in boys are significantly different then both the coefficients of the parenting predictors in girls, for example.

My questions are as follows:
Does it make sense to do this (statistically, I have a theoretical basis)?
How would I do such a thing? I cannot find anything online or in the forum regarding this topic - but it could just be a failure of search terms since I don't know what its called.

I suppose it is too simple to average these coefficients and their s.e's and then still use the formula above?

Thanks in advance for your help,