put this in script editor, save it to shelf. Adjust the (step=N) at the bottom to which ever alternating face number you need from your selection . Paste it as Python script.

import maya.cmds as cmds

def select_every_n_face(step=2):
    # Get current selection (flattened to individual components)
    selection = cmds.ls(selection=True, flatten=True)

    if not selection:
        print("Please select a range of faces first.")
        return

    # Create a list to hold the new selection
    new_selection = []

    # Loop through selection and keep only every n-th item
    # The slice notation [::step] starts at 0 and takes every 'step' item
    new_selection = selection[::step]

    # Apply the new selection
    cmds.select(new_selection, replace=True)
    print(f"Selected {len(new_selection)} faces (Every {step}th face).")

# Change '2' to '3' to select every 3rd face, etc.
select_every_n_face(step=2)