Text this: On co-H-maps to the suspension of the projective plane

  _____      ___      _  __   __   __    _  _   
 /  ___||   / _ \\   | |/ //  \ \\/ //  | \| || 
| // __    / //\ \\  | ' //    \ ` //   |  ' || 
| \\_\ || |  ___  || | . \\     | ||    | .  || 
 \____//  |_||  |_|| |_|\_\\    |_||    |_|\_|| 
  `---`   `-`   `-`  `-` --`    `-`'    `-` -`