Thanks all for the input.

I agree, making that multielement antenna without any means for
measurements is taking a risk that it will be quite inferior to a pro
antenna and thus a waste of time. I found coaxial multielement
antennas at procomm.dk. Length is between 70 cm to 535 cm.

So I back down to a hopefully more humble project. That would be
taking a commercial mobile antenna wip of a 1/4 WL plus !/2 WL
(Smarteq.com). It has a phasing coil inbetween. There is a side mount
with a coaxial FME connector which could be used for a coaxial antenna
with a quarterwave sleeve. The protruding wing (stopper) of the side
mount would be taken off. It is then posible to slide a 1/4 WL Cu tube
over the antenna mount. By fastening the tube to mount ground, one
would obtain the sleeve needed for a coaxial exended antenna. Gain of
the whip in a ground palne confguration is 3 dB.

Should a sleeve +protruding part of mount be eaxtly a quarter
wavelength long at a velocity factor of 1 - or 0.95 ?