Since purely resistive 50 ohm mobile quarter wave antenna doesn't really exist, how are you getting 50 ohms to the transmitter with the quarter wave impedance Transformer ?
Are you using a tuner or some other matching method?
A 1/4 wave length transmission line will invert the antenna impedance a d not repeat it so using a 1/4 wavelength line is wrong to begin with. Secondly a 50 ohm mobile antenne certainly does exist. That's what the base coils are for....impedance matching. In the case of a 1/4 wave antenna the radiaton impedance PLUS ground losses can add to 50 ohms. In fact most people think that is the ultimate natch for a 1/4 wave mobile antenna when in reality it is not. In the end it is quite possible to have a mobile antenna that measures a perfect 50 ohms at the feedpoint.