Israel fired more retaliatory strikes on Gaza on Thursday, in return for rocket attacks which a militant group that associates itself with the Islamic State has claimed responsibility for.

The bombs were aimed at three militant training camps belonging to Hamas and a smaller Palestinian group called Islamic Jihad, reported Reuters, and caused damage but no casualties.

Three Grad rockets were fired into southern Israel on Wednesday evening, the second attack inside a week. In a statement posted online shortly afterwards, a group calling itself the Sheikh Omar Hadid Brigade said it was "launching a battle against the Jews, the enemies of God" and promised further attacks on Israel — but also accused the de-facto rulers of Gaza, Hamas, with "shedding the blood of Muslims."

The group uses Islamic State imagery and music in its videos, and its name is a reference to Omar Hadid, who was a leading al Qaeda figure in the Iraqi insurgency, and whose name was used by the Islamic State of Iraq in 2006, according to analyst Aymenn Jawad Al-Tamimi.

On Tuesday the Sheikh Omar Hadid Brigade also claimed responsibility for firing two rockets that landed around 20 miles across the border into Israel — saying the next attack "would be worse" — and issued a 48-hour ultimatum for Hamas to end a crackdown against supporters of Salafi Muslim groups in Gaza.

Although Hamas has effectively controlled Gaza since it seized power in 2007, internal power struggles have emerged following last summer's war there, which killed around 2,200 Palestinians, mainly civilians, and 73 Israelis, mainly soldiers.

In May a report by the World Bank said the Gaza Strip's economy was on the "verge of collapse" following the failure of post-war reconstruction and the continuing military blockade by Israel and Egypt. Around 40 percent of Gazans live in poverty and around 80 percent are dependent on some sort of aid.

"The overall situation in Gaza… is very difficult right now," said Ehud Ya'ari, a political analyst with the Washington Institute for Near East Policy. "There is open resentment by many sectors of the civilian population against what's going on."

Wednesday's cross-border attack by the brigade came the same day as security forces killed a member of the Salafi militant group in a shootout outside his home in northern Gaza City.

Hamas said that the house of 27-year-old Younis al-Honnor was booby trapped and contained "explosive belts, explosive devises, and rifle-propelled grenades" — but the Sheikh Omar Hadid Brigade labeled it murder committed in "cold blood."

Over the last month Hamas has arrested dozens of members of extremist Salafi groups in the last month and bulldozed a mosque belonging to a group called the "Supporters of the Islamic State in Jerusalem." In retaliation for the demolition of the holy site, the hardline group claimed responsibility for killing a senior Hamas official who it accused of working for an "apostate" government.

Neither of the Salafi groups appear to have yet formerly sworn allegiance to the Islamic State despite claiming their affiliation to the hardline group that has seized large swathes of territory in Iraq and Syria.

In response to both of this week's rocket attacks Israel launched airstrikes against Hamas targets, reflecting its longstanding position of holding Hamas responsible for any rockets launched from inside Gaza, although it acknowledged "rogue elements" were likely responsible for the strikes.

No casualties were incurred on either side but Israeli Defense Minister Moshe Yaalon vowed to "strike harder" if more rockets were fired.

"The situation in Gaza now is that small factions armed with limited quantities of missiles are challenging Hamas policy of maintaining the cease-fire," said Ya'ari. "It's a pattern, by now, in which Israel has to absorb rockets because somebody in Gaza has differences with Hamas."