You can take a look at TGM-Plugin-Activation plugin. It should give you good starting point. As written in the documentation:
TGM Plugin Activation is a PHP library that allows you to easily
require or recommend plugins for your WordPress themes (and plugins).
It allows your users to install and even automatically activate
plugins in singular or bulk fashion using native WordPress classes,
functions and interfaces. You can reference pre-packaged plugins,
plugins from the WordPress Plugin Repository or even plugins hosted
elsewhere on the internet.
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solved Plugin to install a plugin