  [...] The aim of this paper is to study the precision and theoretical biases
in the age determinations of halo stars adopting both asteroseismic and classic
observational constraints. [...] We adopt the well-tested SCEPtER pipeline,
covering evolutionary phases up to the red giant branch (RGB). The fitting
grids contain stars with mass in the range of [0.7; 1.0] $M_{\odot}$ and
metallicity [Fe/H] from -2.5 to -0.5. We investigate several scenarios
characterised by different adopted observational uncertainties. We also assess
the impact of systematic discrepancies between the recovery grid models and
target stars by computing several synthetic grids of stellar models with
perturbed input physics. In our reference scenario, we recover ages for stars
in the main sequence (MS) or subgiant branch (SGB) with a typical 10\%--20\%
precision, while we recover those of RGB stars with a precision of about 60\%.
However, adopting tighter constraints on asteroseismic parameters, the age
precision in RGB improved to 20\%, while few modifications occur in the other
analysed evolutionary phases. A systematic discrepancies between grid models
and target stars shows that a mismatch in the mixing-length parameter value
leads to significant bias in the age estimations for MS stars (about 10\%), but
this bias is smaller for SGB and RGB stars. Neglecting the microscopic
diffusion effect in the recovery grid leads to a typical 40\% bias in age
estimates for stars on the MS. Finally, we applied the technique to stars in
globular clusters. We find a precision in age estimates of around 20\% for MS
stars and up to 40\% for RGB stars, greater than those obtained with classical
methods. We demonstrate the method on stars of the cluster M4, obtaining a
cluster age of $11.9 \pm 1.5$ Gyr and a mass at the turn-of off $0.86 \pm 0.04$
$M_{\odot}$, which are in good agreement with literature results.
