Pictures emerged of the woman the Russian FSB claimed was the killer of Darya Dugina, 29.
The suspect was named as Natalia Shaban-Vovk, 43, mother of a 12-year-old girl.
They are from Mariupol, and the mother is accused of being a member of Ukrainian Azov battalion.
She was allegedly aiming to kill the daughter rather than the father, according to the Russian media.
If Estonia does not extradite Dugina to Russia, there are grounds for tough action against Tallinn, said Senator Vladimir Dzhabarov.
The FSB claimed that Shaban-Vovk had rented an apartment in the same building where Dugina lived, and planned the operation over a number of weeks from her arrival in Russia on July 23.
To monitor Dugina, she used a Mini Cooper, while entering the Russian Federation, DPR number plates were used.
In Moscow, Kazakh plates were used, and when leaving for Estonian, a Ukrainian plate was used.
Russian propagandist Margarita Simonyan, head of RT, urged Putin’s undercover agents to track her down in Estonia.
‘Dasha’s [Darya’s] killers are already in Estonia,” she said.
“Estonia will not - of course - extradite.
“I think we have enough professionals who would want to admire spires around Tallinn.”
Her post included an admission that GRU military intelligence agents operated in Salisbury against Sergei Skripal, who was poisoned with novichok.
The agents were said by Russia - which has always denied an operation to kill Skripal - to be tourists who went to see the beauty of Salisbury Cathedral.
She suggested a killer squad should admire the spires in the Estonian capital.
In his first comment since his daughter’s death, Putin’s ‘Rasputin’ Alexander Dugin, 60, demanded “more than than just revenge or retribution” for her death.
He insisted on victory against Ukraine in the war.
“As a result of a terrorist attack carried out by the Nazi Ukrainian regime, on August 20, when returning from the festival near Moscow, my daughter Darya Dugina was brutally killed by an explosion in front of my eyes.
“She was a beautiful Orthodox girl, a patriot, a military correspondent, an expert on the central channels, and a philosopher.
“Her speeches and reporting have always been profound, grounded and restrained.
“She never called for violence and war.
“She was a rising star at the beginning of her journey. The enemies of Russia meanly, surreptitiously killed her ...
“But we, and our people cannot be broken even by such unbearable blows.
“They wanted to suppress will with bloody terror against the best and most vulnerable of us.
"But they will not succeed.
“Our hearts yearn for more than just revenge or retribution.
“It's too petty, it’s not Russian style.
“We need only our Victory. My daughter laid her life on its altar - so please win!
“We wanted to raise her to be smart, and to be a hero.
“Let this (her tragic death) inspire the sons of our Fatherland to the feat even now.”
