  We consider the Ising model on a supercritical Galton-Watson tree
$\mathbf{T}_n$ of depth $n$ with a sparse random external field, given by a
collection of i.i.d. Bernouilli random variables with vanishing parameter
$p_n$. This may me viewed as a toy model for the Ising model on a configuration
model with a few interfering external vertices carrying a plus spin: the
question is to know how many (or how few) interfering vertices are enough to
influence the whole graph. Our main result consists in providing a necessary
and sufficient condition on the parameters $(p_n)_{n\geq 0}$ for the root of
$\mathbf{T}_n$ to remain magnetized in the large $n$ limit. Our model is
closely related to the Ising model on a (random) pruned sub-tree
$\mathbf{T}_n^*$ with plus boundary condition; one key result is that this
pruned tree turns out to be an inhomogeneous, $n$-dependent, Branching Process.
We then use standard tools such as tree recursions and non-linear capacities to
study the Ising model on this sequence of Galton-Watson trees; one difficulty
is that the offspring distributions of $\mathbf{T}_n^*$, in addition to vary
along the generations $0\leq k \leq n-1$, also depend on~$n$.
