The temperature-pressure phase diagram of the ferromagnet LaCrGe$_3$ is determined for the first time from a combination of magnetization, muon-spin-rotation and electrical resistivity measurements. The ferromagnetic phase is suppressed near $2.1$~GPa, but quantum criticality is avoided by the appearance of a magnetic phase, likely modulated, AFM$_Q$. Our density functional theory total energy calculations suggest a near degeneracy of antiferromagnetic states with small magnetic wave vectors $Q$ allowing for the potential of an ordering wave vector evolving from $Q=0$ to finite $Q$, as expected from the most recent theories on ferromagnetic quantum criticality. Our findings show that LaCrGe$_3$ is a very simple example to study this scenario of avoided ferromagnetic quantum criticality and will inspire further study on this material and other itinerant ferromagnets.
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