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.
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