  New neutral vector bosons $Z'$ decaying to charged gauge boson pairs $W^+W^-$
are predicted in many scenarios of new physics, including models with an
extended gauge sector such as $E_6$, left-right symmetric $Z^\prime_{\rm LRS}$
and the sequential standard model $Z'_{\rm SSM}$. For these benchmark models we
calculate and present theoretical expectations for different values of the
$Z^\prime$ mass $M_2$ and mixing parameter $\xi$. Our results are based on the
narrow width approximation which allows to make a convenient comparison of
experiment to theoretical benchmark models. The diboson production allows to
place stringent constraints on the $Z$-$Z'$ mixing angle and the $Z'$ mass,
which we determine by using data from $pp$ collisions at $\sqrt{s}=13$ TeV
recorded by the ATLAS detector at the CERN LHC, with integrated luminosity of
$\sim$ 36 fb$^{-1}$. By comparing the experimental limits to the theoretical
predictions for the total cross section of $Z'$ resonant production and its
subsequent decay into $W^+W^-$ pairs, we show that the derived constraints on
the mixing angle for the benchmark models are of the order of a few $\times
10^{-4}$, i.e., greatly improved with respect to those derived from the global
analysis of electroweak data. We combine the limits derived from diboson
production data with those obtained from the Drell--Yan process in order to
significantly extend the exclusion region in the $M_{2}$-$\xi$ parameter plane.
Also, we demonstrate that further improvement on the constraining of this
mixing can be achieved through analysis of the full set of Run~II data.
