  We present observations of escaping Lyman Continuum (LyC) radiation from 34
massive star-forming galaxies and 12 weak AGN with reliably measured
spectroscopic redshifts at $z$$\simeq$2.3-4.1. We analyzed Hubble Space
Telescope Wide Field Camera 3 (WFC3) mosaics of the Early Release Science field
in three UVIS filters to sample the rest-frame LyC over this redshift range.
With our best current assessment of the WFC3 systematics, we provide 1$\sigma$
upper limits for the average LyC emission of galaxies at $z$=2.35, 2.75, and
3.60 to $\sim$28.5, 28.1, and 30.7 mag in image stacks of 11-15 galaxies in the
WFC3/UVIS F225W, F275W, and F336W, respectively. The LyC flux of weak AGN at
$z$=2.62 and 3.32 are detected at 28.3 and 27.4 mag with SNRs of $\sim$2.7 and
2.5 in F275W and F336W for stacks of 7 and 3 AGN, respectively, while AGN at
$z$=2.37 are constrained to $\gtrsim$27.9 mag at 1$\sigma$ in a stack of 2 AGN.
The stacked AGN LyC light profiles are flatter than their corresponding
non-ionizing UV continuum profiles out to radii of r$\lesssim$0."9, which may
indicate a radial dependence of porosity in the ISM. With synthetic stellar
SEDs fit to UV continuum measurements longwards of Ly$\alpha$ and IGM
transmission models, we constrain the absolute LyC escape fractions to $f_{\rm
esc}^{\rm abs}$$\simeq$$22^{+44}_{-22}$% at $z$=2.35 and $\lesssim$55% at
$z$=2.75 and 3.60, respectively. All available data for galaxies, including
published work, suggests a more sudden increase of $f_{\rm esc}$ with redshift
at $z$$\simeq$2. Dust accumulating in (massive) galaxies over cosmic time
correlates with increased HI column density, which may lead to reducing $f_{\rm
esc}$ more suddenly at $z$$\lesssim$2. This may suggest that star-forming
galaxies collectively contributed to maintaining cosmic reionization at
redshifts $z$$\gtrsim$2-4, while AGN likely dominated reionization at
$z$$\lesssim$2.
