  The $E0$ transition strength in the $2^+_2 \rightarrow 2^+_1$ transitions of
$^{58,60,62}$Ni have been determined for the first time following a series of
measurements at the Australian National University (ANU) and the University of
Kentucky (UK). The CAESAR Compton-suppressed HPGe array and the Super-e
solenoid at ANU were used to measure the $\delta(E2/M1)$ mixing ratio and
internal conversion coefficient of each transition following inelastic proton
scattering. Level half-lives, $\delta(E2/M1)$ mixing ratios and $\gamma$-ray
branching ratios were measured at UK following inelastic neutron scattering.
The new spectroscopic information was used to determine the $E0$ strengths.
These are the first $2^+ \rightarrow 2^+$ $E0$ transition strengths measured in
nuclei with spherical ground states and the $E0$ component is found to be
unexpectedly large; in fact, these are amongst the largest $E0$ transition
strengths in medium and heavy nuclei reported to date.
