Excited states in $^{58,60,62}$Ni were populated via inelastic proton scattering at the Australian National University as well as via inelastic neutron scattering at the University of Kentucky Accelerator Laboratory. The Super-e electron spectrometer and the CAESAR Compton-suppressed HPGe array were used in complementary experiments to measure conversion coefficients and $delta(E2/M1)$ mixing ratios, respectively, for a number of $2^+ rightarrow 2^+$ transitions. The data obtained were combined with lifetimes and branching ratios to determine $E0$, $M1$, and $E2$ transition strengths between $2^+$ states. The $E0$ transition strengths between $0^+$ states were measured using internal conversion electron spectroscopy and compare well to previous results from internal pair formation spectroscopy. The $E0$ transition strengths between the lowest-lying $2^+$ states were found to be consistently large for the isotopes studied.
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