Known limitations of the Standard Model and the matter-antimatter asymmetry in our universe motivate searches for a violation of the combined symmetry of charge conjugation, parity transformation, and time reversal (CPT). A measurement of the ground-state antihydrogen hyperfine structure has the potential to yield the most sensitive CPT test on an absolute energy scale. At the antiproton decelerator of CERN the ASACUSA-Cusp collaboration progresses towards the first field-free measurement of the hyperfine splitting by applying Rabi-type magnetic resonance spectroscopy to a beam of antihydrogen. High precision and accuracy of this approach have been verified in a measurement of the hydrogen hyperfine splitting using the spectroscopy apparatus built for antihydrogen experiments. The transition F,MF: 1,0 → 0,0 was measured at different external B-field strengths and extrapolated to the zero-field value. At a relative precision of 2.7 ppm the final result is an order of magnitude more precise than earlier in-beam measurements of the hydrogen hyperfine structure. Furthermore it agrees within error bars with the more precise literature value, which is based on maser measurements. Subsequently upgrades have been installed to access the transition F,MF: 1,1 → 0,0 as well. This opportunity is of great practical and theoretical value. Different methods can be used to find the zero-field hyperfine splitting, which are partly less beamtime intensive. Moreover it becomes possible to constrain certain coefficients of the so-called Standard Model Extension framework, some of which don't even require a comparison to antihydrogen. In this talk the status of the antihydrogen hyperfine structure measurement of the ASACUSA collaboration will be presented with a focus on the spectroscopy method and the recent results obtained with hydrogen.