I think I found the bug.

The mainclass HtmlModule implements the Interface IActionable.That means that there is a property

 Public ReadOnly Property ModuleActions() As Entities.Modules.Actions.ModuleActionCollection Implements Entities.Modules.IActionable.ModuleActions
            Get
                Dim Actions As New Entities.Modules.Actions.ModuleActionCollection
                Actions.Add(GetNextActionID, Localization.GetString(Entities.Modules.Actions.ModuleActionType.AddContent, LocalResourceFile), Entities.Modules.Actions.ModuleActionType.AddContent, "", "", EditUrl(), False, Security.SecurityAccessLevel.Edit, True, False)
                Return Actions
            End Get
        End Property

The code is correct. But if  caching is enabled at module level this property is not called from the core. It's only called when an admin views the page. Very strange. This error was recorded in Gemini one year ago !!!!
The workaround I found for my portal is to turn off the caching for the module.

Greetings
J.Sommer