= Solution
Showing that <acceleration> is unchanged by a uniform boost is necessary, but it is not enough to establish the <Galilean covariance of Newton's second law>. One must specify how time, positions, masses, forces and any background physical systems transform, and show that the actual force laws retain their form.
For a constant spatial rotation $R$, boost velocity $\mathbf U$ and translations, write
$$
t'=t+b,\qquad \mathbf x'=R\mathbf x-\mathbf U t+\mathbf a.
$$
Differentiation gives $\mathbf v'=R\mathbf v-\mathbf U$ and $\mathbf a'=R\mathbf a$. Assuming invariant inertial <mass>,
$$
\frac{d\mathbf p'}{dt'}=R\frac{d\mathbf p}{dt}.
$$
Consequently the equation has the same form if the resultant physical <force> satisfies $\mathbf F'=R\mathbf F$. For a pure boost, this reduces to $\mathbf F'=\mathbf F$. More explicitly, if a force is represented by a function of positions, velocities and other physical data, those arguments and data must transform so that its primed value is $R$ times the old value. It is this condition on the complete dynamical model, not the typography of $\mathbf F=m\mathbf a$, that establishes covariance.
An instantaneous inverse-square interaction depending on the separation of two particles passes this test: their common boost cancels from the separation, and a rotation rotates the force vector. A resistance law $\mathbf F=-\zeta(\mathbf v-\mathbf u)$ also passes when the medium's <velocity> $\mathbf u$ transforms with the body. In contrast, keeping $\mathbf F=-\zeta\mathbf v$ in every boosted frame while silently resetting the same medium to rest changes the physical problem. That model has not been shown covariant by noting $\mathbf a'=\mathbf a$.
Likewise, the full <Maxwell equations> are not made Galilean covariant merely because some equation of particle motion is written as force equals mass times acceleration. The covariance of a framework has to be checked together with its interaction laws. Newton's own Corollary V expresses the corresponding equivalence of uniformly translating, nonrotating systems for the mechanics under consideration.<Philosophiae Naturalis Principia Mathematica by Isaac Newton>[1687, Corollary V following the Laws of Motion; the modern boost calculation above makes its dynamical conditions explicit]{ref}
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