Can I merge multiple linear regressions into one regression? I have a dataset of 48,000 records
These records are divided equally into 4 groups
Blue Group, Yellow Group, Green Group, and Red Group
Each one of these groups has 12,000 records
My study has resulted in 4 regression models

Model 1 equation was $a_1 x + b_1$
Model 1 equation was $a_2 x + b_2$
Model 1 equation was $a_3 x + b_3$
Model 1 equation was $a_4 x + b_4$
The thing is that the dataset is not available anymore and I want the regression model for all dataset points
Can I get the average of all 4 models to get the regression for all data points?
So it will be $((a_1 + a_2 + a_3 + a_4)/4)  x + (b_1 + b_2 + b_3 + b_4)/4$
is this model the right model for all data points?

If you have the covariance matrix of the estimated model parameters, too, then you effectively have - what are from a certain perspective - the sufficient statistics. What do I mean with "from a certain perspective"? I mean that you need to assume that each dataset is governed by an intercept + slope model.
You can then describe the model fits e.g. via a normal-inverse-gamma sampling distribution and fit some suitable model on top (either assuming some parameters are the same and/or e.g. vary according to random effects between the different colors). This will then reflect that amongst other things the uncertainty about parameters will differ between separate models depending on the variation in the data and number of data points.
