In classical elections, voters only submit their ballot once, whereas in iterative voting, the ballots may be changed iteratively. Following the work by Wilczynski [20], we consider the case where a social network represents an underlying structure between the voters, meaning that each voter can see her neighbors' ballots. In addition, there is a polling agency, which publicly announces the result for the initial vote. This paper investigates the manipulative power of the polling agency. Previously, Wilczynski [20] studied constructive manipulation for the plurality rule. We introduce destructive manipulation and extend the study to the veto rule. Several restricted variants are considered with respect to their parameterized complexity. The theoretical results are complemented by experiments using different heuristics.