A search is presented for a heavy spin-1 resonance $\mathrm {Z}'$ decaying to a top quark and a vector-like top quark partner $\text {T} $ in the lepton + jets final state. The search is performed using a data set of $\mathrm {p}$ $\mathrm {p}$ collisions at a centre-of-mass energy of 13 $,\text {TeV}$ corresponding to an integrated luminosity of $35.9{,\text {fb}^{-1}} $ as recorded by the CMS experiment at the CERN LHC in the year 2016. The analysis is optimised for final states arising from the $\text {T} $ decay modes to a top quark and a Higgs or $\mathrm {Z}$ boson ( $\text {T} \rightarrow \mathrm {H} \mathrm {t}, \mathrm {Z} \mathrm {t}$ ). The event selection makes use of resolved and merged top quark decay products, as well as decays of boosted Higgs bosons and $\mathrm {Z}$ and $\mathrm {W}$ bosons using jet substructure techniques. No significant deviation from the standard model background expectation is observed. Exclusion limits on the product of the cross section and branching fraction for $\mathrm {Z}'\rightarrow \mathrm {t}\text {T}, \text {T} \rightarrow \mathrm {H} \mathrm {t}, \mathrm {Z} \mathrm {t}, \mathrm {W}\mathrm {b}$ are presented for various combinations of the $\mathrm {Z}'$ resonance mass and the vector-like $\text {T} $ quark mass. These results represent the most stringent limits to date for the decay mode $\mathrm {Z}'\rightarrow \mathrm {t}\text {T} \rightarrow \mathrm {t}\mathrm {H} \mathrm {t}$ . In a benchmark model with extra dimensions, invoking a heavy spin-1 resonance $\mathrm {G}^* $ , masses of the $\mathrm {G}^* $ between 1.5 and 2.3 $,\text {TeV}$ and between 2.0 and 2.4 $,\text {TeV}$ are excluded for $\text {T} $ masses of 1.2 and 1.5 $,\text {TeV}$ , respectively.