| A 120 volt outlet needs one hot, one neutral, and one ground wire. You will not be able to extend from your 240 volt outlet to a 120 volt outlet because the neutral wire is missing.
As you know, all you have at the 240 volt outlet is two hots and the ground.
Please do not consider connecting the neutral of the 120 volt outlet to the ground wire. A neutral wire is considered a current carrying conductor, and the ground wire is reserved for fault currents in the event of short circuits. And if the ground wire is used as a neutral, normal currents are now directed into the entire grounding system, and many possible problems can occur from this connection.
Your only recourse is to run a new cable back to a true 120 volt source. | 
Linear Mode
