A newly fabricated Josephson field effect transistor (JOFET) is coupled with a two-dimensional electron gas (2DEG) in a strained InAs quantum well inserted into an inverted modulation-doped structure with an HEMT-type gate. We indicate the improved characteristics of the JOFET with the HEMT-type gate, instead of the MIS-type gate. The superconducting critical current as well as the junction's normal resistance can be completely controlled via a gate voltage of about -1 V; this provides voltage gain over unity, the first time for a JOFET.