Home or back button as first item in navigation bar?

A bit of background:

I'm in a project where we're building an android system service (meaning it will work regardless of which app is currently in the foreground) for an action bar that will overlay any app and contain some actions/controls. Some of the actions/controls in the bar will include e.g home, back, minimize the bar +3 more.

To my question:

What would be the most logical order for the items in the action bar?

Home feels weird to have as the first item because back is usually located in the top left corner. But having home anywhere else (even as the second item) feels misplaced in relation to the other options.

Any suggestions?

Thanks!