The β-decay of the even-even nucleus $^{70}$Kr with $Z=N+2$, has been investigated at the Radioactive Ion Beam Factory (RIBF) of the RIKEN Nishina Center using the BigRIPS fragment separator, the ZeroDegree Spectrometer, the WAS3ABI implantation station and the EURICA HPGe cluster array. Fifteen γ-rays associated with the β-decay of $^{70}$Kr into $^{70}$Br have been identified for the first time, defining ten populated states below $E_{\mathrm{exc}}=3300$ keV. The half-life of $^{70}$Kr was derived with increased precision and found to be $t_{1/2}=45.19\pm 0.14$ ms. The β-delayed proton emission probability has also been determined as ${\epsilon }_p=0.545\left(23\right)\text{\%}$. An increase in the β-strength to the yrast 1$^{+}$ state in comparison with the heaviest $Z=N+2$ system studied so far ($^{62}$Ge decay) is observed that may indicate increased np correlations in the $T=0$ channel. The β-decay strength deduced from the results is interpreted in terms of the proton-neutron quasiparticle random-phase approximation (pnQRPA) and also with a schematic model that includes isoscalar and isovector pairing in addition to quadrupole deformation. The application of this last model indicates an approximate realization of pseudo-SU(4) symmetry in this system.