Look at how they ask their questions about him:

Q.58 As you may know, Israel had an election earlier this month. The party receiving the third largest number of seats is led by Avigdor Lieberman who has previously called for the execution of Arab members of Israel’s parliament who met with Hamas and whose main campaign message called for Arab citizens of Israel to sign a loyalty oath to the Jewish state in order to prevent their citizenship from being revoked. Do you support or oppose these positions?
...
Q.59 If Avigdor Lieberman becomes a senior member of the Israeli cabinet and refuses to change his positions on Arab citizens of Israel, how would this affect your feelings toward Israel?
- Strengthens my support for Israel against the international criticism that will follow his cabinet appointment
- Weakens my personal connection to Israel because Lieberman’s positions go against my core values
- Has no impact on my feelings toward Israel

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With a depiction like that, it’s surprising that only 69% oppose him.

Now for the second part. I can support him even though this will increase international criticism or I can oppose him due to core values or I can say no impact. Quite a nudge to the poll-taker no?

Also, is it just me, or is the “core values” bit a great setup for a sympathetic newspaper. Here’s how Ha’aretz used it:
...In addition, 32 percent of respondents (40 percent of those younger than 30 years of age) said Lieberman’s appointment to a senior position in the cabinet will weaken their personal link to Israel, as his positions “go against their core values.” ...

It makes it seem as if the readers gave this objection to their values themselves. =)

I wonder how results would change if J Street:
- Quoted him saying that MP’s who help Israel’s enemies are more dangerous than enemy leaders and should be treated as such (instead of using a the term “executions” with little context)
- Mentioned his support for a two state solution and changing the border to reflect demographic realities
- Said that part of his platform included denying state benefits to those who oppose Israel as a Jewish state, refuse civil service, or support the destruction of Israel (instead of using a term like “loyalty oath” and saying it was only for Arabs--something which simply isn’t true)

Unfortunately, a large fraction of the people surveyed and their families and friends--who may not have heard of Avigdor Lieberman before and may have supported some of his pragmatic policies--will now be biased against him and against the Israeli government. This is thanks to a poll from a group that claims to support Israel.