If the button in question is defined in a view (and not generic like the create/edit button in the nav bar), you can easily edit the XML for views in Odoo debug mode. However, I'm not sure if this will permanently solve your problem.

Steps:

• enable debug mode

• go to Settings -> Technical -> User Interface -> Views

• Find the view where the button is located, e.g. search for all views for the model in question and review the views

• If the <button> is found, you can add groups="<your_group>"

For a start, you might review the view product.template.common.form, and have a look at "Configure Variants" on the top with groups="product.group_product_variant". This button is only visible for users, that are allowed to administer product variants.