  NGC 2419 is a peculiar Galactic globular cluster in terms of size/luminosity,
and chemical abundance anomalies. Here, we present Str\"omgren $uvby$
photometry of the cluster. Using the gravity- and metallicity-sensitive $c_1$
and $m_1$ indices, we identify a sample of likely cluster members extending
well beyond the formal tidal radius with an estimated contamination by
non-members of only 1%. We derive photometric [Fe/H] of red giants, and
depending on which literature metallicity relation we use, find reasonable to
excellent agreement with spectroscopic [Fe/H]. We demonstrate explicitly that
the photometric errors are not Gaussian, and using a realistic model for the
photometric uncertainties, find a formal internal [Fe/H] spread of
$\sigma=0.11^{+0.02}_{-0.01}$ dex. This is an upper limit to the cluster's true
[Fe/H] spread and may partially/entirely reflect the limited precision of the
photometric metallicity estimation and systematic effects. The lack of
correlation between spectroscopic and photometric [Fe/H] of individual stars is
further evidence against a [Fe/H] spread on the 0.1 dex level. Finally, the
CN-sensitive $\delta_4$ anti-correlates strongly with Mg abundance, indicating
that the 2nd generation stars are N-enriched. Absence of similar correlations
in some other CN-sensitive indices supports the second generation being
He-rich, which in these indices approximately compensates the shift due to CN.
Compared to a single continuous distribution with finite dispersion, the
observed $\delta_4$ distribution is slightly better fit by two discrete
populations, with the N-enhanced stars accounting for 53$\pm$5%. NGC 2419
appears to be very similar to other metal-poor Galactic globular clusters with
a similarly N-enhanced second generation and little or no variation in [Fe/H],
which sets it apart from other suspected accreted nuclei such as {\omega}Cen.
(abridged)
