= Solution
$Y(s)=\int_0^\infty e^{-st}y(t)dt$, with transforms $sY-y(0)$ and $s^2Y-sy(0)-\dot y(0)$; $e^{\lambda t}$ transforms to $(s-\lambda)^{-1}$ and $\delta$ to $1$. Algebra gives $Y=G H$ with $H=F+(s+2\gamma)y(0)+\dot y(0)$, so $y=g*h$. Writing $a=\sqrt{|\gamma^2-\omega^2|}$ gives $g=e^{-\gamma t}\sinh(at)/a$ for $\gamma\gt \omega$, $e^{-\gamma t}\sin(at)/a$ for $\gamma\lt \omega$, and $te^{-\gamma t}$ at equality. If $y(0)=0$, $h=f+\dot y(0)\delta$, so the initial <velocity> contributes $\dot y(0)g(t)$.
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