Save between steps

Good point, I'm looking for something similar, lets see if someone can bring some light to the matter.

You can do this by overriding or adding a new function to NextStep, example:

In CustomNextStep function add the new logic for updating entity.

/**
 * Implements hook_field_group_form_process().
 */
function my_module_field_group_form_process(array &$element, &$group, &$complete_form): void {
  // Change the submit function of the "next" step.
  if ($next_btn = &$complete_form['actions']['next']) {
    $next_btn['#submit'][0] = [CustomMultiStepManager::class, 'CustomNextStep'];
  }
}

Another quick and better way, which will be added to the module in the future, is to add the "Save" function for the "Next Step" button.

'#submit' => ['simple_multistep_register_next_step', '::save'],

I am upload a patch for this while it is validated by the community and added to a future release.

The patch works for me, but you need to add it in both "showNextButton" and "showBackButton".

I can see the save working with the #4 patch but when I get to the final step, after clicking the node Save button, I get "Error message: The content has either been modified by another user, or you have already submitted modifications. As a result, your changes cannot be saved." This is when adding a new node so I'm guessing there's a check on whether this node is new or not. That would fail since the node is already saved.