adding twig variable in html.html.twig from module
Just tested what I can of your code, and this works from a custom module:

function examplemodule_preprocess_html(&amp;$variables) {

 $variables['cta_bar'] = ['#markup' =&gt; '&lt;h2&gt;Test CTA Bar&lt;/h2&gt;'];
}


It then can be output in the theme with this template:

/themes/custom/mycustomtheme/templates/layout/html.html.twig

like this:

{{ cta_bar }}


Of course, you need to register your preprocess function by clearing the cache.

Other than that, I think it must be a problem with your theme function insusite_cta_bar_actions.  Can you give us more info on that?