  A subset $S$ of a group $(G,+)$ is $t$-weakly sequenceable if there is an
ordering $(y_1, \ldots, y_k)$ of its elements such that the partial sums~$s_0,
s_1, \ldots, s_k$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i y_j$ for $1 \leq
i \leq k$, satisfy $s_i \neq s_j$ whenever and $1 \leq |i-j|\leq t$.
  In [10] it was proved that if the order of a group is $pe$ then all
sufficiently large subsets of the non-identity elements are $t$-weakly
sequenceable when $p > 3$ is prime, $e \leq 3$ and $t \leq 6$. Inspired by this
result, we show that, if $G$ is the semidirect product of $\mathbb{Z}_p$ and
$\mathbb{Z}_2$ and the subset $S$ is balanced, then $S$ admits, regardless of
its size, an alternating parity $t$-weak sequencing whenever $p > 3$ is prime
and $t \leq 8$. A subset of $G$ is balanced if it contains the same number of
even elements and odd elements and an alternating parity ordering alternates
even and odd elements.
  Then using a hybrid approach that combines both Ramsey theory and the
probabilistic method we also prove, for groups $G$ that are semidirect products
of a generic (non necessarily abelian) group $N$ and $\mathbb{Z}_2$, that all
sufficiently large balanced subsets of the non-identity elements admit an
alternating parity $t$-weak sequencing.
  The same procedure works also for studying the weak sequenceability for
generic sufficiently large (not necessarily balanced) sets. Here we have been
able to prove that, if the size of a subset $S$ of a group $G$ is large enough
and if $S$ does not contain $0$, then $S$ is $t$-weakly sequenceable.
